Split (1)

train · 113k rows

text stringlengths 490 29.8k	markdown stringlengths 489 28.6k	html stringlengths 444 61.7M	file_name stringlengths 10 13	ocr stringlengths 0 11.3k
14. Adjuk meg az x2 + y2 −z2 = 1 egyenletű egyköpenyű forgáshiperboloid egy paraméteres egyenletét és határozzuk meg minden pontjában a normálvektorát. Megoldás. r(u, v) = cosh u cos vi + cosh u sin vj + sinh uj ∂r ∂u × ∂r ∂v ∂r ∂v = (sinh u cos vi + sinh u sin vj + cosh uk) × (−cosh u sin vi + cosh u cos vj) = −cosh2 u cos vi −cosh2 u sin vj + cosh u sinh uk 15. Forgassuk meg az y = f(x) függvény graﬁkonját az x tengely körül. Határozzuk meg a kapott felület egy paraméterezését. Megoldás. r(t, φ) = ti + f(t) cos φj + f(t) sin φk, ahol t ∈R, φ ∈[0, 2π]. 16. Egy egyenes körhenger tengelyének két végpontja A = (2, 2, 7) és B = (−1, 5, 3). Adjuk meg a henger palástjának egy paraméterezését, ha a henger sugara 3. Megoldás. A két végpontot összekötő (irány)vektor v = −3i + 3j −4k. A hengernek erre merőleges síkmetszetei körök, a paraméterezéshez a legegyszerűbb, ha keresünk két v-re és egymásra is merőleges egységvektort (egyet könnyű találni, a harmadik vektoriális szorzással előállítható). Ilyenek pl. u = 1 √ 2(i + j) és w = 1 √ 17(−2i + 2j −3k). Ezzel a szorzással előállítható). Ilyenek pl. u = 1 √ 2(i + j) és w = 1 √ 17(−2i + 2j −3k). Ezzel a paraméterezés 1 √ 2(i + j) és w = r(φ, t) = a + tv + 3 cos φu + 3 sin φw, ahol a = 2i + 2j + 7k az A pont helyvektora, φ ∈[0, 2π] és t ∈[0, 1]. 17. Tekintsük azt a tóruszt, amelynek középköre R sugarú, az x−y síkban fekszik, középpontja az origó, és amelynek a keresztmetszete r < R sugarú kör. Adjuk meg a tórusz felületének egy paraméterezését. Megoldás. r(ϑ, ϕ) = (R + r sin ϑ) cos ϕi + (R + r sin ϑ) sin ϕj + r cos ϑk. Egy lehetséges paramétertartomány (ϑ, ϕ) ∈[0, 2π] × [0, 2π]. 18. Az alábbi egyenlőtlenségekkel megadott térrészekhez válasszunk olyan koordinátarendszert, amelyre nézve a paramétertartomány téglatest és határozzuk is meg azt. y p y a) 2 ≤x2 + y2 + z2 ≤4, y ≤0 b) (x −3)2 + (y + 2)2 + (z −1)2 ≤25 ) ( ) (y ) ( ) c) x2 + y2 + z6 ≤1, x ≥0, y ≥0 d) (x −z)2 + 4y2 ≤4, 0 ≤z ≤1 Megoldás. a) gömbi koordinátákkal: r(r, ϑ, ϕ) = r sin ϑ cos ϕi + r sin ϑ sin ϕj + r cos ϑk, (r, ϑ, ϕ) ∈ √ [ 2, 2] × [0, π] × [0, π]. b) eltolt gömbi koordinátákkal: r(r, ϑ, ϕ) = (3+r sin ϑ cos ϕ)i+(−2+r sin ϑ sin ϕ)j+(1+ r cos ϑ)k, (r, ϑ, ϕ) ∈[0, 5] × [0, π] × [0, 2π]. c) az x −y síkban polárkoordinátákat használhatunk, a sík egy pontja feletti egyenessza- kaszt pedig konstans határokkal paraméterezhetjük, pl. r(ρ, φ, h) = ρ cos φi+ρ sin φj+ 6√1 −ρ2hk, (ρ, φ, h) ∈[0, 1] × [0, π/2] × [−1, 1]. d) induljuk ki a szokásos hengerkoordinátákból, de y irányban felére összenyomva, x irány- ban pedig a z koordinátával arányosan eltolva: r(ρ, φ, h) = (h+ρ cos φ)i+ 1 2ρ sin φj+hk, (ρ, φ, h) ∈[0, 2] × [0, 2π] × [0, 1].	14. Adjuk meg az x[2] + y[2] _−_ _z[2]_ = 1 egyenletű egyköpenyű forgáshiperboloid egy paraméteres egyenletét és határozzuk meg minden pontjában a normálvektorát. _Megoldás. r(u, v) = cosh u cos vi + cosh u sin vj + sinh uj_ _∂r_ _∂u_ _[×][ ∂]∂v[r]_ [= (sinh][ u][ cos][ v][i][ + sinh][ u][ sin][ v][j][ + cosh][ u][k][)][ ×][ (][−] [cosh][ u][ sin][ v][i][ + cosh][ u][ cos][ v][j][)] = − cosh[2] _u cos vi −_ cosh[2] _u sin vj + cosh u sinh uk_ 15. Forgassuk meg az y = f (x) függvény grafikonját az x tengely körül. Határozzuk meg a kapott felület egy paraméterezését. _Megoldás. r(t, φ) = ti + f_ (t) cos φj + f (t) sin φk, ahol t ∈, φ ∈ [0, 2π]. R 16. Egy egyenes körhenger tengelyének két végpontja A = (2, 2, 7) és B = (−1, 5, 3). Adjuk meg a henger palástjának egy paraméterezését, ha a henger sugara 3. _Megoldás. A két végpontot összekötő (irány)vektor v = −3i + 3j −_ 4k. A hengernek erre merőleges síkmetszetei körök, a paraméterezéshez a legegyszerűbb, ha keresünk két v-re és egymásra is merőleges egységvektort (egyet könnyű találni, a harmadik vektoriális szorzással előállítható). Ilyenek pl. u = _√1_ _√1_ 2 [(][i][ +][ j][) és][ w][ =] 17 [(][−][2][i][ + 2][j][ −] [3][k][). Ezzel a] paraméterezés r(φ, t) = a + tv + 3 cos φu + 3 sin φw, ahol a = 2i + 2j + 7k az A pont helyvektora, φ ∈ [0, 2π] és t ∈ [0, 1]. 17. Tekintsük azt a tóruszt, amelynek középköre R sugarú, az x _−_ _y síkban fekszik, középpontja_ az origó, és amelynek a keresztmetszete r < R sugarú kör. Adjuk meg a tórusz felületének egy paraméterezését. _Megoldás. r(ϑ, ϕ) = (R + r sin ϑ) cos ϕi + (R + r sin ϑ) sin ϕj + r cos ϑk. Egy lehetséges_ paramétertartomány (ϑ, ϕ) ∈ [0, 2π] × [0, 2π]. 18. Az alábbi egyenlőtlenségekkel megadott térrészekhez válasszunk olyan koordinátarendszert, amelyre nézve a paramétertartomány téglatest és határozzuk is meg azt. a) 2 ≤ _x[2]_ + y[2] + z[2] _≤_ 4, y ≤ 0 b) (x − 3)[2] + (y + 2)[2] + (z − 1)[2] _≤_ 25 c) x[2] + y[2] + z[6] _≤_ 1, x ≥ 0, y ≥ 0 d) (x − _z)[2]_ + 4y[2] _≤_ 4, 0 ≤ _z ≤_ 1 _Megoldás._ a) gömbi koordinátákkal: r(r, ϑ, ϕ) = r sin ϑ cos ϕi + r sin ϑ sin ϕj + r cos ϑk, (r, ϑ, ϕ) ∈ _√_ [ 2, 2] × [0, π] × [0, π]. b) eltolt gömbi koordinátákkal: r(r, ϑ, ϕ) = (3+ _r sin ϑ cos ϕ)i_ +(−2+ _r sin ϑ sin ϕ)j_ +(1+ _r cos ϑ)k, (r, ϑ, ϕ) ∈_ [0, 5] × [0, π] × [0, 2π]. c) az x − _y síkban polárkoordinátákat használhatunk, a sík egy pontja feletti egyenessza-_ kaszt pedig konstans határokkal paraméterezhetjük, pl. r(ρ, φ, h) = ρ cos φi + _ρ sin φj_ + _[√]6_ 1 − _ρ2hk, (ρ, φ, h) ∈_ [0, 1] × [0, π/2] × [−1, 1]. d) induljuk ki a szokásos hengerkoordinátákból, de y irányban felére összenyomva, x irányban pedig a z koordinátával arányosan eltolva: r(ρ, φ, h) = (h+ρ cos φ)i+ [1] 2 _[ρ][ sin][ φ][j][+][h][k][,]_ (ρ, φ, h) ∈ [0, 2] × [0, 2π] × [0, 1]. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">14. Adjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű egyköpenyű forgáshiperboloid egy paraméteres</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletét és határozzuk meg minden pontjában a normálvektorát.</span></p> <p style="top:93.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:117.2pt;left:110.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:133.5pt;left:109.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:133.5pt;left:140.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:147.8pt;left:157.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:174.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15. Forgassuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény graﬁkonját az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely körül. Határozzuk meg a</span></p> <p style="top:188.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kapott felület egy paraméterezését.</span></p> <p style="top:207.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t, φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:227.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16. Egy egyenes körhenger tengelyének két végpontja</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 7)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Adjuk</span></p> <p style="top:241.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">meg a henger palástjának egy paraméterezését, ha a henger sugara</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:261.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A két végpontot összekötő (irány)vektor</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:261.2pt;left:472.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A hengernek</span></p> <p style="top:275.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">erre merőleges síkmetszetei körök, a paraméterezéshez a legegyszerűbb, ha keresünk két</span></p> <p style="top:290.1pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re és egymásra is merőleges egységvektort (egyet könnyű találni, a harmadik vektoriális</span></p> <p style="top:304.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szorzással előállítható). Ilyenek pl.</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:303.0pt;left:298.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:306.1pt;left:295.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:312.7pt;left:302.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:303.0pt;left:395.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:306.1pt;left:390.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:312.7pt;left:397.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">17</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ezzel a</span></sup></p> <p style="top:319.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paraméterezés</span></p> <p style="top:345.4pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ, t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3 cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3 sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">w</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:371.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 7</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pont helyvektora,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:391.2pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">17. Tekintsük azt a tóruszt, amelynek középköre</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú, az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban fekszik, középpontja</span></p> <p style="top:405.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az origó, és amelynek a keresztmetszete</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r < R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör. Adjuk meg a tórusz felületének</span></p> <p style="top:420.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egy paraméterezését.</span></p> <p style="top:439.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Egy lehetséges</span></p> <p style="top:454.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paramétertartomány</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:473.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">18. Az alábbi egyenlőtlenségekkel megadott térrészekhez válasszunk olyan koordinátarendszert,</span></p> <p style="top:487.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amelyre nézve a paramétertartomány téglatest és határozzuk is meg azt.</span></p> <p style="top:502.3pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:518.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">25</span></p> <p style="top:535.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:551.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:571.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:585.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) gömbi koordinátákkal:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i></p> <p style="top:600.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span></p> <p style="top:590.2pt;left:101.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:600.0pt;left:110.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:616.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) eltolt gömbi koordinátákkal:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (3+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(1+</span></p> <p style="top:630.8pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 5]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:647.3pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban polárkoordinátákat használhatunk, a sík egy pontja feletti egyenessza-</span></p> <p style="top:661.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kaszt pedig konstans határokkal paraméterezhetjük, pl.</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ, φ, h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ρ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:676.3pt;left:98.0pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">6</span><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ, φ, h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:692.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) induljuk ki a szokásos hengerkoordinátákból, de</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> irányban felére összenyomva,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> irány-</span></p> <p style="top:707.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ban pedig a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> koordinátával arányosan eltolva:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ, φ, h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:714.4pt;left:470.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:721.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ, φ, h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_242.png	" Megoldás. ríu, 2) vosh t cos vi 4. cosh usin j 4 sinhaj 0r 0r D.. Gsinh a cos e 4. sinh ar sín ej -- cosh a) x (— cosh ar sín el 4. cosh u cosej) 15. Forgassuk meg az y — /(r) függvény grafikonját az 2 tengely körül. Határozzuk meg a kapott felület egy paraméterezését. . Megoldás. rít. ó) — ti 4. f(t) cosój 4. f(t)sin ók, ahol t € R. ó € [9. 27. 16. Egy egyenes körhenger tengelyének két végpontja A — (2,2.7) és B — (-1,5,3). Adjuk meg a henger palástjának egy paramóterezését, ha a henger sugara 3. Megoldás, A két végpontot összekötő Girányjvektot v — —31 - 3j — 4k. A hengermek szorzással előállítható). Ilyenek pl. u — A( 4-3) és w terezéshez a legegyszerűbb, ha keresűnk két s ex találni, a harmadik vektoriális A(Z283.2)— 349. Ezzel a rlé.t) — a 4 fv.4.3c05 u 4 3sinów, ahol a 214-2j-- 7k az A pont helyvektora, ó € [.2z] és ! € [1. 11. ek középköre A sugarú, az 2— y síkban fekszik, középpontja és amelynek a keresztmetszete r 2 AR sugarú kör. Adjuk meg a tórusz felületének egy paraméterezését. 17. Tekintsűk azt a tóruszt, ame .Megoldás. (0,5e) — (R 4-rsind)cos l 4 (R 4 rsind)sinsej 4 reosúk. Egy lehetséges paramétertartomány (9. e) € [0.27] x 0. 21. 18. Az alábbi egyenlőtlenségekkel me DEÉSEVEEPTVET] ) (r ályá 2a (e 25 DEZTALÉTEETVET] ÖE ETTÉTETI adott térrészekhez válasszunk olyan koordinátarendszert a) gömbi koordinátákkak rí.4.g) — rsindcoszi 4 rsindsinyj 4 reosdk, (r.0.e) € IV2.21 x 10. x [. 71. 3) eltolt gömbi koordinátákkal: r(r, 9. ) — (3-Frsindcosg)i4(—24rsindsn j- 14 reosdjk, (r.9.4) € [0,51 x 1071 x 0.271. kaszt pedig konstans határokkal paraméterezhetjük, pl. r(p.6.h) — peosói-: psinój 4 $T1—p3hk, (p.6.h) € (0.1] x [0.2/2] x [-1. 11 ban pedig a : koordínátával arányosan eltola: r(p..h) — (h4-peosó)i4: fpsinéj--Ak, (p.6.A) € [0,2) x [0.2x] x 0. 11.
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 2. feladatsor: Potenciálfüggvény, alakzatok paraméterezése (megoldás) 1. Potenciálos-e az alábbi vektormező? Ha igen, adjuk meg egy potenciálját. a) u(x, y) = yi + xj b) u(x, y, z) = zex+sin yi + zex+sin y cos yj + ex+sin yk ) ( , y, ) yj c) u(x, y, z) = (x2 + yz)i + (y −x2)j + (z + xy)k Megoldás. Mindhárom vektormező az egész térben (síkon) értelmezett, tehát akkor pontenciálos, ha a rotációja 0. ∂ ∂ , a) ∂x ∂ , ∂x ∂x −∂y ∂y ∂y ∂y = 1 −1 = 0, tehát potenciálos. Egy potenciál Z x Z y 0 uy(0, η) dη f(x, y) = Z y Z 0 ux(ξ, y) dξ + 0 Z x Z y Z 0 y dξ + Zy 0 0 dη = xy b) ∂ux ∂ ∂ux ∂ ∂ux ∂ zex+sin(y) z cos(y)ex+sin(y) ex+sin(y) ∂x ∂y ∂z ∂uy ∂uy ∂uy ∂x ∂ y ∂uy ∂y ∂ ∂uy  ze z cos(y)e e z cos(y)ex+sin(y) z cos2(y)ex+sin(y) −z sin(y)ex+sin(y) cos(y)ex+sin(y)   ∂x ∂y ∂z ∂uz ∂uz ∂uz ∂x ∂ ∂x y ∂uz ∂y ∂ ∂y ∂uz ∂z = z ex+sin(y) cos(y)ex+sin(y)  szimmetrikus, tehát a vektormező potenciálos. Egy potenciál f(x, y, z) = Z x Z y Z z Z 0 ux(ξ, y, z) dξ + Zy 0 uy(0, η, z) dη + Z 0 uz(0, 0, ζ) dζ 0 Z x Z y 0 Z z Z x 0 zeξ+sin y dξ + Z y 0 zesin η cos η dη + Z 0 1 dζ 0 0 0 = (ex −1)zesin y + (esin y −1)z + z = zex+sin y c) y −x2) ∂x = −2x ̸= z = ∂(x2 + yz) ∂y ∂(y −x2) ∂y tehát ez a vektormező nem potenciálos. (rot u = xi −(2x + z)k) 2. Centrális vektormezőnek nevezzük a v(r) = f(\|r\|) r \|r\| alakú vektormező Centrális vektormezőnek nevezzük a v(r) = f(\|r\|) r \|r\| alakú vektormezőket, ahol f : R+ →R tetszőleges diﬀerenciálható függvény. Mutassuk meg, hogy minden centrális vektormező potenciálos és határozzuk meg egy potenciálfüggvényt. Megoldás. A szimmetria miatt elég a deriváltmátrix egy főátlón kívüli elemét számolni, pl. az első komponens y szerinti deriváltja a láncszabály alapján: ∂y p y f(√x2 + y2 + z2) √x2 + y2 + z2 x2 + y2 + z2 j y pj f ′(√x2 + y2 + z2) √x2 + y2 + z2 −f(√x2 + y2 + z2) x2 + y2 + z2 xy √x2 + y2 + z2, ami x és y cseréjére szimmetrikus, tehát a rotáció esetleg az origó kivételével mindenhol 0. v az origó körüli forgatásokra nézve szimmetrikus, így sejthetjük, hogy a potenciálfüggvény is ilyen (ha létezik). Számoljuk ki F(√x2 + y2 + z2) gradiensét, ahol F tetszőleges függvény: grad F( q x2 + y2 + z2) = F ′( q x2 + y2 + z2) xi + yj + zk √x2 + y2 + z2, tehát ha F ′ = f, akkor a gradiens éppen v. Mivel f folytonos, ilyen tulajdonságú F függvény létezik (primitív függvény).	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 2. feladatsor: Potenciálfüggvény, alakzatok paraméterezése (megoldás) 1. Potenciálos-e az alábbi vektormező? Ha igen, adjuk meg egy potenciálját. a) u(x, y) = yi + xj b) u(x, y, z) = ze[x][+sin][ y]i + ze[x][+sin][ y] cos yj + e[x][+sin][ y]k c) u(x, y, z) = (x[2] + yz)i + (y − _x[2])j + (z + xy)k_ _Megoldás. Mindhárom vektormező az egész térben (síkon) értelmezett, tehát akkor ponten-_ ciálos, ha a rotációja 0. a) _[∂x]_ _∂x_ _[−]_ _[∂y]∂y_ [= 1][ −] [1 = 0, tehát potenciálos. Egy potenciál] � _x_ � _y_ _f_ (x, y) = 0 _[u][x][(][ξ, y][) d][ξ][ +]_ 0 _[u][y][(0][, η][) d][η]_ � _x_ � _y_ = 0 _[y][ d][ξ][ +]_ 0 [0 d][η][ =][ xy] b)   c) _∂ux_ _∂ux_ _∂ux_ _∂x_ _∂y_ _∂z_ _∂uy_ _∂uy_ _∂uy_ _∂x_ _∂y_ _∂z_ _∂uz_ _∂uz_ _∂uz_ _∂x_ _∂y_ _∂z_   =  _ze[x][+sin(][y][)]_ _z cos(y)e[x][+sin(][y][)]_ _e[x][+sin(][y][)]_ _z cos(y)e[x][+sin(][y][)]_ _z cos[2](y)e[x][+sin(][y][)]_ _−_ _z sin(y)e[x][+sin(][y][)]_ cos(y)e[x][+sin(][y][)]  _e[x][+sin(][y][)]_ cos(y)e[x][+sin(][y][)] 0   szimmetrikus, tehát a vektormező potenciálos. Egy potenciál � _x_ � _y_ � _z_ _f_ (x, y, z) = 0 _[u][x][(][ξ, y, z][) d][ξ][ +]_ 0 _[u][y][(0][, η, z][) d][η][ +]_ 0 _[u][z][(0][,][ 0][, ζ][) d][ζ]_ � _x_ � _y_ � _z_ = 0 _[ze][ξ][+sin][ y][ d][ξ][ +]_ 0 _[ze][sin][ η][ cos][ η][ d][η][ +]_ 0 [1 d][ζ] = (e[x] _−_ 1)ze[sin][ y] + (e[sin][ y] _−_ 1)z + z = ze[x][+sin][ y] _∂(y −_ _x[2])_ = −2x ̸= z = _[∂][(][x][2][ +][ yz][)],_ _∂x_ _∂y_ tehát ez a vektormező nem potenciálos. (rot u = xi − (2x + z)k) 2. Centrális vektormezőnek nevezzük a v(r) = f (\|r\|) [r] _\|r\|_ [alakú vektormezőket, ahol][ f][ :][ R][+][ →] [R] tetszőleges differenciálható függvény. Mutassuk meg, hogy minden centrális vektormező potenciálos és határozzuk meg egy potenciálfüggvényt. _Megoldás. A szimmetria miatt elég a deriváltmátrix egy főátlón kívüli elemét számolni, pl._ az első komponens y szerinti deriváltja a láncszabály alapján: _√_ _√_ _∂_ �f ( _x2 + y2 + z2)_ � �f ′( _x2 + y2 + z2)_ � _xy_ _√_ _x_ = _√_ _−_ _[f]_ [(][√][x][2][ +][ y][2][ +][ z][2][)] _√_ _,_ _∂y_ _x2 + y2 + z2_ _x2 + y2 + z2_ _x[2]_ + y[2] + z[2] _x2 + y2 + z2_ ami x és y cseréjére szimmetrikus, tehát a rotáció esetleg az origó kivételével mindenhol 0. v az origó körüli forgatásokra nézve szimmetrikus, így sejthetjük, hogy a potenciálfüggvény is ilyen (ha létezik). Számoljuk ki F ([√]x[2] + y[2] + z[2]) gradiensét, ahol F tetszőleges függvény: grad F (�x[2] + y[2] + z[2]) = F _[′](�_ _x[2]_ + y[2] + z[2])√[x][i][ +][ y][j][ +][ z][k] _,_ 2 2 2 _x_ + y + z tehát ha F _[′]_ = f, akkor a gradiens éppen v. Mivel f folytonos, ilyen tulajdonságú F függvény létezik (primitív függvény). -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">2. feladatsor: Potenciálfüggvény, alakzatok</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">paraméterezése (megoldás)</span></b></p> <p style="top:141.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Potenciálos-e az alábbi vektormező? Ha igen, adjuk meg egy potenciálját.</span></p> <p style="top:155.8pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:172.2pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ze</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ze</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:188.6pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:205.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Mindhárom vektormező az egész térben (síkon) értelmezett, tehát akkor ponten-</span></p> <p style="top:220.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciálos, ha a rotációja</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:239.8pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></sup></p> <p style="top:248.0pt;left:98.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></sup></p> <p style="top:248.0pt;left:129.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát potenciálos. Egy potenciál</span></sup></p> <p style="top:271.8pt;left:128.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:259.4pt;left:178.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:283.6pt;left:184.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ, y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:259.4pt;left:262.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:283.6pt;left:268.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup></p> <p style="top:298.7pt;left:166.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:286.3pt;left:178.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:310.5pt;left:184.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:286.3pt;left:230.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:310.5pt;left:236.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i></sup></p> <p style="top:323.5pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span></p> <p style="top:336.2pt;left:85.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:353.7pt;left:85.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:342.4pt;left:93.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></p> <p style="top:351.3pt;left:95.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:342.4pt;left:119.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></p> <p style="top:351.3pt;left:122.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:342.4pt;left:146.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></p> <p style="top:351.3pt;left:149.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂z</span></i></p> <p style="top:358.3pt;left:93.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:368.2pt;left:95.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:358.3pt;left:119.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:368.2pt;left:122.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:358.3pt;left:146.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:368.2pt;left:149.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂z</span></i></p> <p style="top:375.3pt;left:93.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">z</span></i></p> <p style="top:384.1pt;left:95.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:375.3pt;left:120.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">z</span></i></p> <p style="top:384.1pt;left:122.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:375.3pt;left:146.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">z</span></i></p> <p style="top:384.1pt;left:149.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂z</span></i></p> <p style="top:336.2pt;left:162.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:353.7pt;left:162.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:339.2pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:356.7pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:346.4pt;left:207.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ze</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:346.4pt;left:326.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:346.4pt;left:477.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:361.1pt;left:191.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:361.1pt;left:279.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:361.1pt;left:462.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:375.7pt;left:210.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:375.7pt;left:330.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:375.7pt;left:494.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:339.2pt;left:531.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:356.7pt;left:531.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:397.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szimmetrikus, tehát a vektormező potenciálos. Egy potenciál</span></p> <p style="top:421.8pt;left:128.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:409.4pt;left:190.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:433.6pt;left:195.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ, y, z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:409.4pt;left:285.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:433.6pt;left:290.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, η, z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:409.4pt;left:380.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:433.6pt;left:386.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:448.7pt;left:177.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:436.2pt;left:190.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:460.5pt;left:195.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ze</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:436.2pt;left:274.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:460.5pt;left:280.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ze</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:436.2pt;left:374.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:460.5pt;left:380.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:471.4pt;left:177.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ze</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ze</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup></p> <p style="top:491.9pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span></p> <p style="top:511.6pt;left:128.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:527.9pt;left:145.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:519.7pt;left:180.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:527.9pt;left:274.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:519.7pt;left:308.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:546.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát ez a vektormező nem potenciálos. (</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:563.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Centrális vektormezőnek nevezzük a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:570.7pt;left:330.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">\|</span></i><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú vektormezőket, ahol</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span></sup><sup><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i></sup><sup><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span></sup></p> <p style="top:577.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tetszőleges diﬀerenciálható függvény. Mutassuk meg, hogy minden centrális vektormező</span></p> <p style="top:592.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">potenciálos és határozzuk meg egy potenciálfüggvényt.</span></p> <p style="top:609.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A szimmetria miatt elég a deriváltmátrix egy főátlón kívüli elemét számolni, pl.</span></p> <p style="top:623.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az első komponens</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szerinti deriváltja a láncszabály alapján:</span></p> <p style="top:643.3pt;left:103.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:659.6pt;left:100.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:635.4pt;left:116.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:650.7pt;left:134.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:651.4pt;left:215.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:635.4pt;left:221.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:651.4pt;left:233.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:635.4pt;left:245.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ′</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:650.7pt;left:264.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:651.4pt;left:349.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:659.6pt;left:375.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:635.4pt;left:451.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:643.3pt;left:491.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i></p> <p style="top:650.7pt;left:462.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:678.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ami</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cseréjére szimmetrikus, tehát a rotáció esetleg az origó kivételével mindenhol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:693.2pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az origó körüli forgatásokra nézve szimmetrikus, így sejthetjük, hogy a potenciálfüggvény</span></p> <p style="top:707.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">is ilyen (ha létezik). Számoljuk ki</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gradiensét, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tetszőleges függvény:</span></p> <p style="top:734.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">grad</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></p> <p style="top:723.2pt;left:145.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:734.4pt;left:154.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></p> <p style="top:723.2pt;left:253.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:734.4pt;left:263.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup></p> <p style="top:733.7pt;left:331.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:761.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor a gradiens éppen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:761.2pt;left:326.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mivel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> folytonos, ilyen tulajdonságú</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvény létezik (primitív függvény).</span></p> </div>	page_243.png	Matematika A3 gyakorlat Energetika és Mechatronika HSc szakok, 2016/17 ősz 2. feladatsor: Potenciálfüggvény, alakzatok Pparaméterezése (megoldás 1. Potenciálos-e az alábbi vektormező? Ha igen, adjuk meg egy potenciálját a) uley) — s j 1) ulry.2) — er 4 zertés cosyj 4 ETtIk ) ulzy.2)— (É 414 [Y j3 (s 4 zv)k " Megoldás. Mindhárom vektormező az egész térben (síkon) értelmezett, tehát akkor ponten- ciálos, ha a rotációja 0. tehát potenciálos. Egy potenciál 1é.h kéé s [uttdn [évágs [/990-9 t a vektormező potenciálos. Egy potenciál [k: zhat 4 [/ap0.nzydn 4 [u0.0.0ac Verz- tehát ez a vektormező nem potenciálos. (rot u — zi — (2r 4 2)k) ezzük a vír) — f(lr)) § alaki ket, ahol / : B, — R iálható függvény. . Mutassuk meg, hogy minden centrális vektormező ektormez határozzuk meg egy potenciálfű Megoldás. A a deriváltmátrix egy főátlón kívüli el 2 [EE ] - ( LTE - LTE ö Á VaE t MYE -ETKEI Er VEE ha létezik). Számoljuk ki F(/777 4777 2) gradiensét, ahol F tetszöleges fűggyé FEr Va tt t számolni, pl. sad FYE FT Z) a FIE TFT Z)
3. Mutassuk meg, hogy u(x, y, z) = x2i + 3xz2j −2xzk vektorpotenciálos és adjuk meg egy vektorpotenciálját. Megoldás. A vektormező mindenhol értelmezett, div u = 2x + 0 −2x = 0, tehát létezik vektorpotenciál. Z z vx(x, y, z) = = vy(x, y, z) = Z 0 uy(x, y, ζ) dζ 0 Z z Zz 0 (3xζ2) dζ = xz3 0 Z x Z z Z 0 uz(ξ, y, 0) dξ − Z 0 ux(x, y, ζ) dζ 0 Z x 0Z z Z 0 (−2 · ξ · 0) dξ − Z z 0 x2 dζ = −x2z Eszerint v(x, y, z) = xz3i −x2zj egy vektorpotenciál. 4. Adjuk meg az alábbi görbék egy paraméterezését: a) A = (2, 1, 5) és B = (−1, 9, 11) pontokat összekötő szakasz b) origó középpontú, a és b hosszúságú, az x ill. y tengelyekkel párhuzamos féltengelyekkel rendelkező ellipszis c) az x2 + y2 + z2 = a2 és x + 2y = 0 egyenletű felületek metszésvonala. Megoldás. a) Legyen a = 2i + j + 5k és b = −i + 9j + 11k a két végpont helyvektora. Ekkor r(t) = a + t(b −a) a szakasz paraméterezése, ha t ∈[0, 1]. b) Az egységkör egy kényelmes paraméterezése cos ti+sin tj (t ∈[0, 2π]). Ebből nyújtással kapunk ellipszist: r(t) = a cos ti + b sin tj. c) A második egyenletből x = −2y, amit az elsőbe írva 5y2 + z2 = a2 adódik. Az x koordináta nélkül ez egy olyan ellipszis, ami az y −z síkban helyezkedik el, a tengelyek szimmetriatengelyei, tehát az előzőek mintájára a √ 5 cos tj+a sin tk egy paraméterezésa. szimmetriatengelyei, tehát az előzőek mintájára a √ 5 cos tj+a sin tk egy paraméterezésa. Az x koordinátát y meghatározza, így a metszésvonal így paraméterezhető: r(t) = −2a √ 5 cos ti + a √ 5 cos tj + a sin tk (t ∈[0, 2π]). a √ 5 cos ti + a √ 5 cos tj + a sin tk (t ∈[0, 2π]). 5. Milyen alakzat paraméterezése az r(t) = R(cos ti + sin tj) + atk, ha R > 0? Megoldás. Az utolsó koordinátát elhagyva az x −y síkban fekvő R sugarú körhöz jutunk, “egyenletesen” paraméterezve. Az utolsó koordináta eközben lineárisan növekszik, tehát az alakzat csavarvonal, ami egy R sugarú henger palástján helyezkedik el. Mivel a vetület 2π szerint periodikus, a menetemelkedés 2πa. 6. Adjuk meg az a = (2, 1, 9), b = (1, 5, 10) és c = (0, 4, 0) helyvektorú pontokat tartalmazó sík egy paraméteres egyenletét és ennek segítségével írjuk fel egy normálvektorát. Megoldás. r(u, v) = a + u(b −a) + v(c −a) = (2 −u −2v)i + (1 + 4u + 3v)j + (9 + u −9v)k ∂r ∂u × ∂r ∂v = (−i + 4j + k) × (−2i + 3j −9k) = −39i −11j + 5k. 7. Tekintsük az origó középpontú egységgömb felszínét a szokásos paraméterezéssel. Adjuk meg az alábbi egyenlőtlenségek által meghatározott daraboknak megfelelő paramétertartományt: a) z ≥0 b) x2 + y2 ≤z2 c) z ≤ 1 √ 2 d) x ≥0	3. Mutassuk meg, hogy u(x, y, z) = x[2]i + 3xz[2]j − 2xzk vektorpotenciálos és adjuk meg egy vektorpotenciálját. _Megoldás. A vektormező mindenhol értelmezett, div u = 2x + 0 −_ 2x = 0, tehát létezik vektorpotenciál. � _z_ _vx(x, y, z) =_ 0 _[u][y][(][x, y, ζ][) d][ζ]_ � _z_ = 0 [(3][xζ] [2][) d][ζ][ =][ xz][3] � _x_ � _z_ _vy(x, y, z) =_ 0 _[u][z][(][ξ, y,][ 0) d][ξ][ −]_ 0 _[u][x][(][x, y, ζ][) d][ζ]_ � _x_ � _z_ = 0 [(][−][2][ ·][ ξ][ ·][ 0) d][ξ][ −] 0 _[x][2][ d][ζ][ =][ −][x][2][z]_ Eszerint v(x, y, z) = xz[3]i − _x[2]zj egy vektorpotenciál._ 4. Adjuk meg az alábbi görbék egy paraméterezését: a) A = (2, 1, 5) és B = (−1, 9, 11) pontokat összekötő szakasz b) origó középpontú, a és b hosszúságú, az x ill. y tengelyekkel párhuzamos féltengelyekkel rendelkező ellipszis c) az x[2] + y[2] + z[2] = a[2] és x + 2y = 0 egyenletű felületek metszésvonala. _Megoldás._ a) Legyen a = 2i + j + 5k és b = −i + 9j + 11k a két végpont helyvektora. Ekkor r(t) = a + t(b − a) a szakasz paraméterezése, ha t ∈ [0, 1]. b) Az egységkör egy kényelmes paraméterezése cos ti+sin tj (t ∈ [0, 2π]). Ebből nyújtással kapunk ellipszist: r(t) = a cos ti + b sin tj. c) A második egyenletből x = −2y, amit az elsőbe írva 5y[2] + z[2] = a[2] adódik. Az x koordináta nélkül ez egy olyan ellipszis, ami az y − _z síkban helyezkedik el, a tengelyek_ szimmetriatengelyei, tehát az előzőek mintájára _√a_ 5 [cos][ t][j] [+] _[a][ sin][ t][k][ egy paraméterezésa.]_ Az x koordinátát y meghatározza, így a metszésvonal így paraméterezhető: r(t) = _−_ _√[2][a]_ _√a_ 5 [cos][ t][i][ +] 5 [cos][ t][j][ +][ a][ sin][ t][k][ (][t][ ∈] [[0][,][ 2][π][]).] 5. Milyen alakzat paraméterezése az r(t) = R(cos ti + sin tj) + atk, ha R > 0? _Megoldás. Az utolsó koordinátát elhagyva az x −_ _y síkban fekvő R sugarú körhöz jutunk,_ “egyenletesen” paraméterezve. Az utolsó koordináta eközben lineárisan növekszik, tehát az alakzat csavarvonal, ami egy R sugarú henger palástján helyezkedik el. Mivel a vetület 2π szerint periodikus, a menetemelkedés 2πa. 6. Adjuk meg az a = (2, 1, 9), b = (1, 5, 10) és c = (0, 4, 0) helyvektorú pontokat tartalmazó sík egy paraméteres egyenletét és ennek segítségével írjuk fel egy normálvektorát. _Megoldás. r(u, v) = a + u(b −_ a) + v(c − a) = (2 − _u −_ 2v)i + (1 + 4u + 3v)j + (9 + u − 9v)k _∂r_ _∂u_ _[×][ ∂]∂v[r]_ [= (][−][i][ + 4][j][ +][ k][)][ ×][ (][−][2][i][ + 3][j][ −] [9][k][) =][ −][39][i][ −] [11][j][ + 5][k][.] 7. Tekintsük az origó középpontú egységgömb felszínét a szokásos paraméterezéssel. Adjuk meg az alábbi egyenlőtlenségek által meghatározott daraboknak megfelelő paramétertartományt: a) z ≥ 0 b) x[2] + y[2] _≤_ _z[2]_ c) z ≤ _√1_ 2 d) x ≥ 0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Mutassuk meg, hogy</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektorpotenciálos és adjuk meg egy</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektorpotenciálját.</span></p> <p style="top:92.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A vektormező mindenhol értelmezett,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát létezik</span></p> <p style="top:107.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektorpotenciál.</span></p> <p style="top:136.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:123.5pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:147.8pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:162.9pt;left:159.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:150.4pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:174.7pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xζ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:189.8pt;left:106.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:177.3pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:201.6pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ, y,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:177.3pt;left:266.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:201.6pt;left:271.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:216.7pt;left:159.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:204.2pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:228.5pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:204.2pt;left:268.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:228.5pt;left:274.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup></p> <p style="top:247.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Eszerint</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egy vektorpotenciál.</span></p> <p style="top:266.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Adjuk meg az alábbi görbék egy paraméterezését:</span></p> <p style="top:280.6pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 5)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 11)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontokat összekötő szakasz</span></p> <p style="top:297.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) origó középpontú,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> hosszúságú, az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ill.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelyekkel párhuzamos féltengelyekkel</span></p> <p style="top:311.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rendelkező ellipszis</span></p> <p style="top:327.9pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű felületek metszésvonala.</span></p> <p style="top:347.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:361.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) Legyen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> b</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 11</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a két végpont helyvektora.</span></p> <p style="top:361.5pt;left:508.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ekkor</span></p> <p style="top:375.9pt;left:97.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a szakasz paraméterezése, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:392.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) Az egységkör egy kényelmes paraméterezése</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">). Ebből nyújtással</span></p> <p style="top:406.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kapunk ellipszist:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:423.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) A második egyenletből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amit az elsőbe írva</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik. Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:437.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">koordináta nélkül ez egy olyan ellipszis, ami az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban helyezkedik el, a tengelyek</span></p> <p style="top:452.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szimmetriatengelyei, tehát az előzőek mintájára</span></p> <p style="top:450.6pt;left:347.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></p> <p style="top:453.7pt;left:344.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:460.3pt;left:351.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egy paraméterezésa.</span></sup></p> <p style="top:468.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> koordinátát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> meghatározza, így a metszésvonal így paraméterezhető:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:482.5pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></sup></p> <p style="top:484.0pt;left:108.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:490.6pt;left:115.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:481.0pt;left:167.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></p> <p style="top:484.0pt;left:163.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:490.6pt;left:170.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">).</span></sup></p> <p style="top:503.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Milyen alakzat paraméterezése az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> at</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:522.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az utolsó koordinátát elhagyva az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban fekvő</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú körhöz jutunk,</span></p> <p style="top:536.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">“egyenletesen” paraméterezve. Az utolsó koordináta eközben lineárisan növekszik, tehát az</span></p> <p style="top:551.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakzat csavarvonal, ami egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú henger palástján helyezkedik el. Mivel a vetület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:565.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szerint periodikus, a menetemelkedés</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πa</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:584.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Adjuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 9)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> b</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 10)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyvektorú pontokat tartalmazó</span></p> <p style="top:599.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sík egy paraméteres egyenletét és ennek segítségével írjuk fel egy normálvektorát.</span></p> <p style="top:618.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (1 + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (9 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:641.8pt;left:108.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:658.1pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:658.1pt;left:138.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">39</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:678.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Tekintsük az origó középpontú egységgömb felszínét a szokásos paraméterezéssel. Adjuk</span></p> <p style="top:693.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">meg az alábbi egyenlőtlenségek által meghatározott daraboknak megfelelő paramétertarto-</span></p> <p style="top:707.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mányt:</span></p> <p style="top:721.9pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:738.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:754.8pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i></p> <p style="top:753.2pt;left:124.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:756.3pt;left:120.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:762.9pt;left:127.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:772.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_244.png	3. Mutassuk meg, hogy u(z. v. vektorpotenciálját. 224 4 3r22j — 2z2k vektorpotenciálos és adjuk meg egy .Megoldás. A vektormező mindenhol vektorpotenciál. rtelmezett, dívü — 2740 -— 2r — 0, ványo)a [/dlry.0A£ [/ozcwac [/94tu.00£- [/tdry.90c [/C2-£-094£- [/4c- - Er Eszerint víz. y.2) egy vektorpotenciál. 4. Adjuk meg az alábbi görbék egy paraméterezését: a) Az (2.1.5) és B — (—1.9.11) pontokat összekötő szakasz 1) origó középpontú, a és b hoss rendelkező ellípszis CEE iságú, az x ll. y tengelyekkel párhuzamos félte és 2 4. 24 — 0 egyenletű felületek metszésvonala. . Megoldás. a) Legyen a — 24-j4-5k és b — -1 4.9j 4 11K a két végpont helyvektora. Ekkor H(t) — a 4 t(b — a) a szakasz paraméterezése, ha ! € [. 11. V) Az egységkör egy kényelmes paraméterezése cos t14-sin tj (£ € [0,22]). Ebből nyújtással kapunk ellipszist: r() — acosti 4 bsin j. e) A második egyenletből x — —2y, amit az elsőbe írva 54. 4 koordínáta né a adódik. Az x cül ez egy olyan ellipszis, ami az y — z síkban helyezkedik el, a tengelyek szímmetriatengelyi tehát az előzőek mintájára - costj4-a sinfk egy paran 24 costi 4 3s costj 4 asíntk (t € [0.2x)). 5. Milyen alakzat paraméterezése az r() — F(costi 4 sintj) 4. atk, ha R. 02 alakzat csavarvonal, ami egy A sugarú henger palástján helyezkedik el. Mivel a vetület 27 szerint periodikus, a menetemelkedés 27a. (2.1.9). b — (1.5.10) és e teres egyenletét és ennek se Hulb-a)telc-a) y paraméterezhető: r(t) — 6. Adjuk meg az sík egy paran (0, 4.0) helyvektorű pontokat tartalmazó p tségével írjuk fel es tormálvektorát. 2—4—2014(14-444-30)-4(94-4—90)k . Megoldás. r(u, e) ör 0r E 7. Tekintsük az origó kö 2 (-14-4j 4.x (—214- 3) — 9k) — —391 — 11j 4. 5k. pontú egységgőmb felszínét a szokásos paraméterezéssel. Mjuk a z20 DEE Öz:s4 CEE
nem szimmetrikus, tehát nincsen potenciál. 10. Legyen f : R+ →R diﬀerenciálható és tekintsük a v(x, y, z) = f(√x2 + y2) xi+yj √ x2+y2 vektor- mezőt. Mutassuk meg, hogy v potenciálos és határozzuk meg egy potenciálfüggvényét. Megoldás. v a z tengely körüli forgatásokra nézve szimmetrikus, így sejthetjük, hogy a potenciálfüggvény is ilyen (ha létezik). Számoljuk ki F(√x2 + y2) gradiensét, ahol F tetszőleges függvény: grad F( q x2 + y2) = F ′( q x2 + y2) xi + yj √x2 + y2, tehát ha F ′ = f, akkor a gradiens éppen v. Mivel f folytonos, ilyen tulajdonságú F függvény létezik. 11. Az origón átmenő tengely körül w ∈R3 szögsebességgel forgó test r helyvektorú pontjának sebessége u(r) = w × r. Határozzuk meg u egy vektorpotenciálját. Megoldás. Koordinátákkal u(x, y, z) = (wyz −wzy)i + (wzx −wxz)j + (wxy −wyx)k. A divergencia számolásához a komponenseket rendre x, y és z szerint kell deriválni. Ezek a deriváltak mind 0-val egyenlőek, tehát div u = 0, vagyis létezik vektorpotenciál. Egy lehetséges vektorpotenciál komponensfüggvényei: tehát ha F ′ = f, akkor a gradiens éppen v. Mivel f folytonos, ilyen tulajdonságú F függvény létezik. 11. Az origón átmenő tengely körül w ∈R3 szögsebességgel forgó test r helyvektorú pontjának ( ) Z z vx(x, y, z) = = vy(x, y, z) = Z 0 uy(x, y, ζ) dζ Z z z2 Z 0 (wzx −wxζ) dζ = wzxz −wx 0 Z x Z z Z 0 uz(ξ, y, 0) dξ − Z 0 ux(x, y, ζ) dζ 0 Z x Z z Z 0 (wxy −wyξ) dξ − Z 0 (wyζ −wzy) dζ = wxxy + wzyz −wy 0 x2 + z2 vz(x, y, z) = 0. 12. Adjuk meg az x2 + y2 = z2 kúpfelület és az x + z = 1 egyenletű sík metszésvonalának egy paraméterezését. Megoldás. A második egyenletből z = 1 −x, ezt az elsőbe írva x2 + y2 = (1 −x)2 = 1 + x2 −2x, tehát 1 −y2 = 2x. Eszerint x és z kifejezhető y segítségével, válasszuk ezt paraméternek: t = y. A paraméterezés r(t) = 1 −t2 2 i + tj + 1 + t2 2 k, ahol t ∈R. 13. Az y2 + z2 = a2 és x2 + z2 = b2 egyenletű hengerfelületek metszésvonala a ̸= b esetén két zárt görbéből áll. Adjuk meg ezek egy-egy paraméterezését az a < b esetben. Megoldás. A két görbe a nagyobb sugarú henger egy-egy oldalán helyezkedik el, tehát az egyik az x < 0, a másik az x > 0 félsíkban. Emiatt megtehetjük, hogy a kisebb sugarú henger yz síkra való vetületét (a sugarú kör) paraméterezzük, és a második egyenletből kifejezzük az x értékeket. Ebből a következő paraméterezések adódnak (t ∈[0, 2π]): r(t) = q b2 −a2 sin2 ti + a cos tj + a sin tk r(t) = − q b2 −a2 sin2 ti + a cos tj + a sin tk.	nem szimmetrikus, tehát nincsen potenciál. 10. Legyen f : R+ → R differenciálható és tekintsük a v(x, y, z) = f ([√]x[2] + y[2])√xxi+[2]+yjy[2][ vektor-] mezőt. Mutassuk meg, hogy v potenciálos és határozzuk meg egy potenciálfüggvényét. _Megoldás. v a z tengely körüli forgatásokra nézve szimmetrikus, így sejthetjük, hogy a_ potenciálfüggvény is ilyen (ha létezik). Számoljuk ki F ([√]x[2] + y[2]) gradiensét, ahol F tetszőleges függvény: grad F (�x[2] + y[2]) = F _[′](�_ _x[2]_ + y[2])√[x][i][ +][ y][j] _,_ 2 2 _x_ + y tehát ha F _[′]_ = f, akkor a gradiens éppen v. Mivel f folytonos, ilyen tulajdonságú F függvény létezik. 11. Az origón átmenő tengely körül w ∈ szögsebességgel forgó test r helyvektorú pontjának R[3] sebessége u(r) = w × r. Határozzuk meg u egy vektorpotenciálját. _Megoldás. Koordinátákkal u(x, y, z) = (wyz −_ _wzy)i + (wzx −_ _wxz)j + (wxy −_ _wyx)k. A_ divergencia számolásához a komponenseket rendre x, y és z szerint kell deriválni. Ezek a deriváltak mind 0-val egyenlőek, tehát div u = 0, vagyis létezik vektorpotenciál. Egy lehetséges vektorpotenciál komponensfüggvényei: � _z_ _vx(x, y, z) =_ 0 _[u][y][(][x, y, ζ][) d][ζ]_ � _z_ _z[2]_ = 0 [(][w][z][x][ −] _[w][x][ζ][) d][ζ][ =][ w][z][xz][ −]_ _[w][x]_ 2 � _x_ � _z_ _vy(x, y, z) =_ 0 _[u][z][(][ξ, y,][ 0) d][ξ][ −]_ 0 _[u][x][(][x, y, ζ][) d][ζ]_ � _x_ � _z_ = 0 [(][w][x][y][ −] _[w][y][ξ][) d][ξ][ −]_ 0 [(][w][y][ζ][ −] _[w][z][y][) d][ζ]_ _x[2]_ + z[2] = wxxy + wzyz − _wy_ 2 _vz(x, y, z) = 0._ 12. Adjuk meg az x[2] + y[2] = z[2] kúpfelület és az x + z = 1 egyenletű sík metszésvonalának egy paraméterezését. _Megoldás. A második egyenletből z = 1 −_ _x, ezt az elsőbe írva_ _x[2]_ + y[2] = (1 − _x)[2]_ = 1 + x[2] _−_ 2x, tehát 1 − _y[2]_ = 2x. Eszerint x és z kifejezhető y segítségével, válasszuk ezt paraméternek: _t = y. A paraméterezés_ r(t) = [1][ −] _[t][2]_ i + tj + [1 +][ t][2] k, 2 2 ahol t ∈ . R 13. Az y[2] + z[2] = a[2] és x[2] + z[2] = b[2] egyenletű hengerfelületek metszésvonala a ̸= b esetén két zárt görbéből áll. Adjuk meg ezek egy-egy paraméterezését az a < b esetben. _Megoldás. A két görbe a nagyobb sugarú henger egy-egy oldalán helyezkedik el, tehát az_ egyik az x < 0, a másik az x > 0 félsíkban. Emiatt megtehetjük, hogy a kisebb sugarú henger yz síkra való vetületét (a sugarú kör) paraméterezzük, és a második egyenletből kifejezzük az x értékeket. Ebből a következő paraméterezések adódnak (t ∈ [0, 2π]): � r(t) = _b[2]_ _−_ _a[2]_ sin[2] _ti + a cos tj + a sin tk_ � r(t) = − _b[2]_ _−_ _a[2]_ sin[2] _ti + a cos tj + a sin tk._ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nem szimmetrikus, tehát nincsen potenciál.</span></p> <p style="top:76.5pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható és tekintsük a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:74.6pt;left:469.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">i</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">j</span></b></p> <p style="top:75.1pt;left:463.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:86.5pt;left:473.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektor-</span></sup></p> <p style="top:95.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mezőt. Mutassuk meg, hogy</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> potenciálos és határozzuk meg egy potenciálfüggvényét.</span></p> <p style="top:113.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely körüli forgatásokra nézve szimmetrikus, így sejthetjük, hogy a</span></p> <p style="top:127.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">potenciálfüggvény is ilyen (ha létezik). Számoljuk ki</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gradiensét, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tet-</span></p> <p style="top:142.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szőleges függvény:</span></p> <p style="top:168.9pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">grad</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></p> <p style="top:157.7pt;left:145.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:168.9pt;left:154.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></p> <p style="top:157.7pt;left:228.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:168.9pt;left:238.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup></p> <p style="top:168.2pt;left:280.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:196.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor a gradiens éppen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:196.8pt;left:326.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mivel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> folytonos, ilyen tulajdonságú</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> F</span></i></p> <p style="top:211.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvény létezik.</span></p> <p style="top:228.7pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Az origón átmenő tengely körül</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szögsebességgel forgó test</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyvektorú pontjának</span></p> <p style="top:243.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sebessége</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Határozzuk meg</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egy vektorpotenciálját.</span></p> <p style="top:260.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Koordinátákkal</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A</span></p> <p style="top:275.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">divergencia számolásához a komponenseket rendre</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szerint kell deriválni. Ezek</span></p> <p style="top:289.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a deriváltak mind</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-val egyenlőek, tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagyis létezik vektorpotenciál. Egy</span></p> <p style="top:303.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lehetséges vektorpotenciál komponensfüggvényei:</span></p> <p style="top:328.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:316.4pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:340.6pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:359.0pt;left:159.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:346.5pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:370.7pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:350.9pt;left:334.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:367.2pt;left:337.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:385.9pt;left:106.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:373.4pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:397.6pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ, y,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:373.4pt;left:266.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:397.6pt;left:271.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, ζ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:412.8pt;left:159.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:400.3pt;left:171.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:424.5pt;left:177.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:400.3pt;left:278.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> z</span></i></p> <p style="top:424.5pt;left:283.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ζ</span></i></sup></p> <p style="top:442.9pt;left:159.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:434.8pt;left:266.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:451.1pt;left:282.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:465.2pt;left:106.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:487.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Adjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kúpfelület és az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű sík metszésvonalának egy</span></p> <p style="top:501.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paraméterezését.</span></p> <p style="top:518.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A második egyenletből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ezt az elsőbe írva</span></p> <p style="top:540.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></p> <p style="top:562.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Eszerint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kifejezhető</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> segítségével, válasszuk ezt paraméternek:</span></p> <p style="top:576.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A paraméterezés</span></p> <p style="top:605.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:613.2pt;left:154.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:605.0pt;left:172.8pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:613.2pt;left:226.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:605.0pt;left:245.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:629.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:647.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13. Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> b</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletű hengerfelületek metszésvonala</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén két</span></p> <p style="top:661.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">zárt görbéből áll. Adjuk meg ezek egy-egy paraméterezését az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a < b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetben.</span></p> <p style="top:679.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A két görbe a nagyobb sugarú henger egy-egy oldalán helyezkedik el, tehát az</span></p> <p style="top:693.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyik az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x <</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a másik az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> félsíkban. Emiatt megtehetjük, hogy a kisebb sugarú</span></p> <p style="top:707.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">henger</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkra való vetületét (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör) paraméterezzük, és a második egyenletből</span></p> <p style="top:722.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kifejezzük az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékeket. Ebből a következő paraméterezések adódnak (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">):</span></p> <p style="top:748.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:736.0pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:748.3pt;left:151.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">b</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:771.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:759.0pt;left:150.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:771.3pt;left:160.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">b</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_246.png	10. Legyen f R -R díllaendálbató és tentsük a víz.9.) a JYETTT l voktor AMegoldás. v. a 2 tengely körüli forgatásokra nézva szimmetrikus, így sejtbet potenciálfüggvény is ilyen (ha létezik). Számoljuk ki F(V377-42) gradiens szöleges függvény: srad F( ) - FIE LA EET: ük, hogy a ahol F tet. tehát ha F" — /, akkor a gradiens éppen v. függvény létezik. 11. Az origón átmenő tengely körül w € R? szögsebességgel forgó test r helyvektorú pontjának bessége ulr) — w x r. Határozzuk meg u egy vektorpotenciálját. "Megoldás. Koordinátákkal u(r.y.2) — (ityz — tegy)i 4. (tesz — t,2)j 4 (tesy — 1yz)k. A divergencia számolásához a komponcnscket rendre z, y és 2 szerint kell deriválni. Ezek a deríváltak mind 0-val egyenlőek, tehát dívu — 0. vagyis létezik vektorpotenciál. Egy lehetséges vektorpotenciól komponcnsfüggvényei: tlo [9 y.04c [es-e9k : 9.046- [ selo0ik [Fe — ) — [7 toyt — tesyk zy 4 182 — 18 v. 12. Adjuk meg az £ 4 4 paraméterezését .Megoldás. A második egyenletből 2 — 1 — 2. ezt az elsőbe írva 129 Er 4at — 22. tehát 1— ? — 2r. Eszerint x és 4. A param kifejezhető y s egítségével, válasszuk ezt paraméternek: Eő] ahol ! € R. 13. Az — 42 egyenletű hengerfelületek metszésvonala a 2 b esetén két zárt görbéből áll. Adjuk meg ezek egy-egy paraméterezését az a 2 6 csetben. .Megoldás. A két görbe a nagyobb sugarú henger egy-egy oldalán helyezkedik el, tel egyik az x 2 0, a másik az x 2. 0 félsíkban. Emiatt megtehetjük, hogy a kisebb susarú henger y2 síkra való vetületét (a sugarú kör) paramóterezzük, és a második egyenletből kifejezzük az 2 értékeket. Ebből a következő paraméterezések adódnak (t € [1. 27]): l — VZ s 4 acostj 4 asíntk eli át az VÍ — etsin?ti 4 acostj 4 asintk.
12. Homogén tömegeloszlású m tömegű vékony drótból a oldalú négyzet alakú keretet hajlítunk. Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatékát. 13. Mi az u(x, y, z) = (y+z)i+(x+z)j+(x+y)k vektormező integrálja az AB szakasz mentén, ha A = (1, −2, 3), B = (2, 1, 4)? 14. Legyen u : R3 →R3 az alábbi vektormező: u(x, y, z) = xyi + (y −z2)j + (2z −xy)k Határozza meg az u integrálját a) az origó középpontú, x-y síkban felvő egységkörvonalon a z tengely pozitív fele felől nézve pozitív körüljárási irányban b) ezen körvonal y ≥0 félkörén az előbbivel megegyező irányban. 15. Integráljuk az u(x, y, z) = (2xy −z)i + (x2 + z)j + (y −x)k vektormezőt az 1 9 ) Integráljuk az u(x, y, z) = (2xy −z)i + (x2 + z)j + (y −x)k vektormezőt az 1 9x2 + 1 16y2 = 1, z = 2 egyenletrendszerrel megadott ellipszisen a z tengely pozitív fele irányából nézve pozitív körüljárás szerint. 1 9x2 + 1 16	12. Homogén tömegeloszlású m tömegű vékony drótból a oldalú négyzet alakú keretet hajlítunk. Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatékát. 13. Mi az u(x, y, z) = (y +z)i+(x+z)j+(x+y)k vektormező integrálja az AB szakasz mentén, ha A = (1, −2, 3), B = (2, 1, 4)? 14. Legyen u : _→_ az alábbi vektormező: R[3] R[3] u(x, y, z) = xyi + (y − _z[2])j + (2z −_ _xy)k_ Határozza meg az u integrálját a) az origó középpontú, x-y síkban felvő egységkörvonalon a z tengely pozitív fele felől nézve pozitív körüljárási irányban b) ezen körvonal y ≥ 0 félkörén az előbbivel megegyező irányban. 15. Integráljuk az u(x, y, z) = (2xy − _z)i + (x[2]_ + z)j + (y − _x)k vektormezőt az_ [1] 9 _[x][2][ +][ 1]16_ _[y][2][ = 1,]_ _z = 2 egyenletrendszerrel megadott ellipszisen a z tengely pozitív fele irányából nézve_ pozitív körüljárás szerint. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Homogén tömegeloszlású</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű vékony drótból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> oldalú négyzet alakú keretet hajlítunk.</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatékát.</span></p> <p style="top:90.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> AB</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szakasz mentén,</span></p> <p style="top:104.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:120.9pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">14. Legyen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">→</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az alábbi vektormező:</span></p> <p style="top:147.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:173.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Határozza meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> integrálját</span></p> <p style="top:188.1pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) az origó középpontú,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban felvő egységkörvonalon a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely pozitív fele felől</span></p> <p style="top:202.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nézve pozitív körüljárási irányban</span></p> <p style="top:219.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) ezen körvonal</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> félkörén az előbbivel megegyező irányban.</span></p> <p style="top:235.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15. Integráljuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:242.7pt;left:462.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:242.7pt;left:493.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:249.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletrendszerrel megadott ellipszisen a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely pozitív fele irányából nézve</span></p> <p style="top:264.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pozitív körüljárás szerint.</span></p> </div>	page_247.png	n tömegeloszlású m tömegű w. yzet alakú keretet hajl Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatókát 13. Miaz ulr.y.2) — (y2jiá (r 23). B-(2.1.497 14. Legyen u : R? — R? az alábbi vektormező: ank. 4 (z4-y)k vektormező integrálja az AB szakasz mentén, ulr.y. zút[y— 2Y 4 (22 — zv)k Határozza meg az u integrálját a) az origó középpontú, 2-y síkban felvő egységkörvonalon a mézve pozítív körüljárási irányban tengely pozitív fele felől 15. Integráljuk az ulr,y,2) — (2ry— 21- 2-- 2j- [y — 2)k vektormezőt az 1424. by? — 1. gyező irányban.
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 3. feladatsor: Görbe ívhossza, görbementi integrál 1. Mi az r(t) = t3 3 √ 5 2j + 9 1. Mi az r(t) = t3 3 i + 6 2 5 t 2t2k görbe ívhossza a t ∈[1, 2] intervallumon? 2. Tekintsük a síkon az t3 3 i + 6 √ 2 5 t 5 2j + 9 2 r(t) = f(t) cos ti + f(t) sin tj paraméteres egyenletű spirált (csigavonal), ahol f diﬀerenciálható függvény. Mekkora a görbe 0 ≤t ≤2π szakaszának (azaz egy körülfordulásnak) ívhossza, ha a) f(t) = t (arkhimédeszi spirál) b) f(t) = αt valamilyen rögzített α > 0 értékkel (logaritmikus spirál) 3. Integráljuk az f(x, y, z) = √1 + 4x + 9yz skalármezőt az r(t) = t2i+tj+t3k görbe mentén t = 0 és t = 1 paraméterértékek között. 4. Egy vékony L hosszúságú drót vonalmenti sűrűsége egyik végétől a másikig lineárisan vál- tozik µ és 2µ között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközéppontja? 5. Mennyi az u(x, y, z) = (y2 −x2)i + 2yzj −x2k vektormező integrálja az r(t) = ti + t2j + t3k görbe mentén t = 0-tól t = 1-ig? 6. Integráljuk az u : R3 →R3 u(x, y, z) = yi + xj + 2zk vektormezőt az r : R →R3 r(t) = 2t+1 −1 i + t 4 + 2t2j + sin π 4 t e(1−t)2k térgörbe mentén a t = 0 és t = 1 paraméterértékeknek megfelelő pontok között. 7. Mi az u(x, y) = −y x2+y2i + x x2+y2j vektormező integrálja a) az origó körüli R sugarú kör mentén pozitív irányítással b) az origó körüli R sugarú kör mentén negatív irányítással c) az (5, 9) pont körüli 2 sugarú kör mentén pozitív irányítással d) az r(t) = αi + tj egyenes mentén (α > 0) t = −∞-től t = +∞-ig? További gyakorló feladatok √ 8. Mi az r(t) = et cos ti + et sin tj + etk görbe ívhossza a t ∈[0, 2] intervallumon? 9. Mi az r(t) = (sinh t + cosh t)i + (cosh t −sinh t)j + 2tk görbe ívhossza a t ∈[0, ln 2] intervallumon? 10. Mennyi az r(t) = 2 sin(t)i + 2 cos(t)j + ! t2 2 −ln t térgörbe 1 ≤t ≤√e paraméterértékeknek megfelelő részének ívhossza? 11. Egy homogén tömegeloszlású vékony kötelet két végénél fogva lógatunk. Az általa meg- határozott görbe egy paraméterezése r(t) = ti + cosh tj, t ∈[−1, 1]. Hol van a kötél tömegközéppontja?	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 3. feladatsor: Görbe ívhossza, görbementi integrál _√_ 1. Mi az r(t) = _[t][3]_ 2 52 j + [9] 3 [i][ +][ 6] 5 _[t]_ 2 _[t][2][k][ görbe ívhossza a][ t][ ∈]_ [[1][,][ 2] intervallumon?] 2. Tekintsük a síkon az r(t) = f (t) cos ti + f (t) sin tj paraméteres egyenletű spirált (csigavonal), ahol f differenciálható függvény. Mekkora a görbe 0 ≤ _t ≤_ 2π szakaszának (azaz egy körülfordulásnak) ívhossza, ha a) f (t) = t (arkhimédeszi spirál) b) f (t) = α[t] valamilyen rögzített α > 0 értékkel (logaritmikus spirál) 3. Integráljuk az f (x, y, z) = _[√]1 + 4x + 9yz skalármezőt az r(t) = t[2]i_ + _tj_ + _t[3]k görbe mentén_ _t = 0 és t = 1 paraméterértékek között._ 4. Egy vékony L hosszúságú drót vonalmenti sűrűsége egyik végétől a másikig lineárisan változik µ és 2µ között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközéppontja? 5. Mennyi az u(x, y, z) = (y[2] _−_ _x[2])i + 2yzj −_ _x[2]k vektormező integrálja az r(t) = ti + t[2]j + t[3]k_ görbe mentén t = 0-tól t = 1-ig? 6. Integráljuk az u : _→_ R[3] R[3] u(x, y, z) = yi + xj + 2zk vektormezőt az r : _→_ R R[3] r(t) = �2[t][+1] _−_ 1� i + _t_ � _π_ � _e[(1][−][t][)][2]k_ 4 + 2t[2] [j][ + sin] 4 _[t]_ térgörbe mentén a t = 0 és t = 1 paraméterértékeknek megfelelő pontok között. 7. Mi az u(x, y) = _−y_ _x_ _x[2]+y[2]_ [i][ +] _x[2]+y[2]_ [j][ vektormező integrálja] a) az origó körüli R sugarú kör mentén pozitív irányítással b) az origó körüli R sugarú kör mentén negatív irányítással c) az (5, 9) pont körüli 2 sugarú kör mentén pozitív irányítással d) az r(t) = αi + tj egyenes mentén (α > 0) t = −∞-től t = +∞-ig? ## További gyakorló feladatok 8. Mi az r(t) = e[t] cos ti + e[t] sin tj + e[t]k görbe ívhossza a t ∈ [0, 2] intervallumon? _√_ 9. Mi az r(t) = (sinh t + cosh t)i + (cosh t − sinh t)j + 2tk görbe ívhossza a t ∈ [0, ln 2] intervallumon? 10. Mennyi az r(t) = 2 sin(t)i + 2 cos(t)j + �t2 � 2 _[−]_ [ln][ t] k térgörbe 1 ≤ _t ≤_ _[√]e paraméterértékeknek megfelelő részének ívhossza?_ 11. Egy homogén tömegeloszlású vékony kötelet két végénél fogva lógatunk. Az általa meghatározott görbe egy paraméterezése r(t) = ti + cosh tj, t ∈ [−1, 1]. Hol van a kötél tömegközéppontja? -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">3. feladatsor: Görbe ívhossza, görbementi integrál</span></b></p> <p style="top:132.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup></p> <p style="top:140.1pt;left:147.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 6</span></sup></p> <p style="top:124.6pt;left:177.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:131.2pt;left:184.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:140.1pt;left:179.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:129.9pt;left:195.7pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">5</span></p> <p style="top:136.2pt;left:195.7pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span></sup></p> <p style="top:140.1pt;left:220.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe ívhossza a</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> intervallumon?</span></sup></p> <p style="top:149.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Tekintsük a síkon az</span></p> <p style="top:174.9pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:200.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paraméteres egyenletű spirált (csigavonal), ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható függvény. Mekkora a</span></p> <p style="top:215.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbe</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szakaszának (azaz egy körülfordulásnak) ívhossza, ha</span></p> <p style="top:229.6pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (arkhimédeszi spirál)</span></p> <p style="top:246.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">valamilyen rögzített</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékkel (logaritmikus spirál)</span></p> <p style="top:262.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Integráljuk az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> skalármezőt az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe mentén</span></p> <p style="top:276.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterértékek között.</span></p> <p style="top:293.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Egy vékony</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> L</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> hosszúságú drót vonalmenti sűrűsége egyik végétől a másikig lineárisan vál-</span></p> <p style="top:307.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tozik</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközép-</span></p> <p style="top:322.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pontja?</span></p> <p style="top:338.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:353.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbe mentén</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-tól</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ig?</span></p> <p style="top:369.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Integráljuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">→</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:395.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:421.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektormezőt az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:448.8pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:438.9pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:448.8pt;left:147.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:438.9pt;left:187.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:448.8pt;left:195.6pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:440.8pt;left:230.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:457.0pt;left:215.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span></sup></p> <p style="top:435.9pt;left:286.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:457.0pt;left:295.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:435.9pt;left:307.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:448.8pt;left:316.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(1</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:478.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">térgörbe mentén a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterértékeknek megfelelő pontok között.</span></p> <p style="top:494.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:493.0pt;left:168.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:502.3pt;left:161.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:493.4pt;left:216.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:502.3pt;left:206.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja</span></sup></p> <p style="top:510.6pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) az origó körüli</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör mentén pozitív irányítással</span></p> <p style="top:527.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) az origó körüli</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör mentén negatív irányítással</span></p> <p style="top:543.5pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 9)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pont körüli</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör mentén pozitív irányítással</span></p> <p style="top:559.9pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenes mentén (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = +</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ig?</span></p> <p style="top:592.5pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:616.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe ívhossza a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> intervallumon?</span></p> <p style="top:633.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:623.3pt;left:354.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:633.1pt;left:364.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe ívhossza a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln 2]</span></p> <p style="top:647.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">intervallumon?</span></p> <p style="top:664.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Mennyi az</span></p> <p style="top:696.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2 sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2 cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:680.6pt;left:250.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:704.7pt;left:260.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup></p> <p style="top:680.6pt;left:300.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:696.5pt;left:309.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:730.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">térgörbe</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterértékeknek megfelelő részének ívhossza?</span></p> <p style="top:746.7pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Egy homogén tömegeloszlású vékony kötelet két végénél fogva lógatunk. Az általa meg-</span></p> <p style="top:761.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">határozott görbe egy paraméterezése</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:761.2pt;left:455.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hol van a kötél</span></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tömegközéppontja?</span></p> </div>	page_248.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 3. feladatsor: Görbe ívhossza, görbementi integrál 1. Mi az r(t) — 514. BÉdj 4. 2Ek görbe ívhossza a ! € [1,2) intervallumon? el0) — fl)eosti 4 f(ősintj görbe 0 £ t £ 2r szakaszának (azaz egy körülfordulásnak) ívhossza, ha. a) /(€) — t (arkhimédeszi spirál) 1) /(0) — a valamilyen rögzített a - 0 értékkel (logarítmikus spirál) 3. Integráljuk az /(r.y.2) — VIGTE 47 skalármezőt az r() — £14-tj4-$k görbe menté 4. Egy vékony L hosszúságűú drót vonalmenti sűrűsége egyik végétől a másikig lincárisan vá tozik ja és 24 között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközép- pontja? 5. Mennyi az u. 4. görbe mentén £ — 0-tól 6. Integráljuk az u : R? — R? ( — a?)i 4-2y2) — 29k vektormező integrálja az r(t) — 4434 Ék 1is ulnyz) í jt 22k vektormezőt az r : R R r -k C" ll tn ( 0ést 1 7. Miaz ulz,y) 2 l3 eszálja paraméterértékeknek megfelelő pontok között. 1) az origó körüli A sugarú kör €) az (5.9) pont körüli 2 sugarú kör mentén pozi d) az r(t) — ai 4 tj egyenes mentén (a — 0) £ — "További gyakorló feladatok $. Mi azr(0) — Écosti 4 esintj 4. €k gőrbe ívhossza a ! € [1,2 intervallmon? 9. Mi az rít) — (sinh! 4 coshí)i 4. (cosht — sinhí)j 4. Vtk görbe fehossza a ! € [.1a2] A 10. M. rtőkeknek megfelelő ré 11. Egy homogén tömegeloszlású vékony kötelet két végénél fogya lógatunk. Az általa n határozott görbe egy paraméterezése e(t) — ti 4 coshtj, £ € [-1,1. Hol van a köt tömegközéppontja? eli 2sin(ti 4 2005(03 rgörbe 1 £ £ £ /7 paraméte k írhossza? " tél
A négy szakaszon \|˙r(t)\| egyaránt a (konstans). A π/2 szögű forgásszimmetria miatt a két tengelyre vonatkozó tehetetlenségi nyomaték megegyezik. Pl. az x tengelyre nézve: Z 1 a2 Z 1 Z 1 Z 1 a2 2 (1 −t)2a dt + a2 a2 2 (−t)2a dt + a2 a2 2 (−1 + t)2a dt a2 Θ = 4a a 2 t2a dt + 4a 4a 4a = ma2 8 Z 1 dt = ma2 8 4t2 −4t + 2 4 · 1 3 1 3 −4 · 1 2 1 2 + 2 = ma2 6 . = ma2 6 13. Mi az u(x, y, z) = (y+z)i+(x+z)j+(x+y)k vektormező integrálja az AB szakasz mentén, ha A = (1, −2, 3), B = (2, 1, 4)? Megoldás. Használjuk az r(t) = (i−2j+3k)+t(i+3j+k) paraméterezést (t ∈[0, 1]), ezzel u · dr = Z 1 Z 0 u(r(t)) · ˙r(t) dt 0 Z 1 Z 0 ((−2 + 3t + 3 + t)i + (1 + t + 3 + t)j + (1 + t −2 + 3t)k) · (i + 3j + k) dt 0 Z 1 Z 0 (12 + 14t) = 19. 14. Legyen u : R3 →R3 az alábbi vektormező: u(x, y, z) = xyi + (y −z2)j + (2z −xy)k Határozza meg az u integrálját a) az origó középpontú, x-y síkban felvő egységkörvonalon a z tengely pozitív fele felől nézve pozitív körüljárási irányban b) ezen körvonal y ≥0 félkörén az előbbivel megegyező irányban. Megoldás. A paraméterezés legyen r(t) = cos ti + sin tj, ekkor a) u · dr = Z 2π (cos t sin ti + sin tj −cos t sin tk) · (−sin ti + cos tj) dt 0 Z 2π cos t sin t −cos t sin2 t dt = 0 b) u · dr = Z π cos t sin t −cos t sin2 t dt = 0. 15. Integráljuk az u(x, y, z) = (2xy −z)i + (x2 + z)j + (y −x)k vektormezőt az 1 9 Integráljuk az u(x, y, z) = (2xy −z)i + (x2 + z)j + (y −x)k vektormezőt az 1 9x2 + 1 16y2 = 1, z = 2 egyenletrendszerrel megadott ellipszisen a z tengely pozitív fele irányából nézve pozitív körüljárás szerint. 1 9x2 + 1 16 á ábó Megoldás. A deriváltmátrix ∂ux ∂ux ∂ux     2y 2x −1 2x 0 1 −1 1 0 ∂x ∂y ∂z ∂uy ∂uy ∂uy ∂x ∂ ∂y ∂uy ∂y ∂ ∂z ∂uy  ∂x ∂y ∂z ∂uz ∂uz ∂uz ∂x ∂ ∂x ∂y ∂uz ∂y ∂ ∂y ∂ ∂uz ∂z =  , ami szimmetrikus, tehát u potenciálos. A megadott görbe zárt, tehát a Newton-Leibniztétel alapján az integrál 0.	A négy szakaszon \|r˙(t)\| egyaránt a (konstans). A π/2 szögű forgásszimmetria miatt a két tengelyre vonatkozó tehetetlenségi nyomaték megegyezik. Pl. az x tengelyre nézve: � 1 _m_ _a[2]_ � 1 _m_ _a[2]_ � 1 _m_ _a[2]_ � 1 _m_ _a[2]_ Θ = 0 4a 2 _[t][2][a][ d][t][ +]_ 0 4a 2 [(1][ −] _[t][)][2][a][ d][t][ +]_ 0 4a 2 [(][−][t][)][2][a][ d][t][ +] 0 4a 2 [(][−][1 +][ t][)][2][a][ d][t] = _[ma][2]_ � 1 �4t[2] _−_ 4t + 2� dt = _[ma][2]_ �4 · [1] � = _[ma][2]_ 8 0 8 3 _[−]_ [4][ ·][ 1]2 [+ 2] 6 _[.]_ 13. Mi az u(x, y, z) = (y +z)i+(x+z)j+(x+y)k vektormező integrálja az AB szakasz mentén, ha A = (1, −2, 3), B = (2, 1, 4)? _Megoldás. Használjuk az r(t) = (i_ _−_ 2j +3k)+ _t(i_ +3j + k) paraméterezést (t ∈ [0, 1]), ezzel � � 1 u · dr = 0 [u][(][r][(][t][))][ ·][ ˙][r][(][t][) d][t] � 1 = 0 [((][−][2 + 3][t][ + 3 +][ t][)][i][ + (1 +][ t][ + 3 +][ t][)][j][ + (1 +][ t][ −] [2 + 3][t][)][k][)][ ·][ (][i][ + 3][j][ +][ k][) d][t] � 1 = 0 [(12 + 14][t][) = 19][.] 14. Legyen u : _→_ az alábbi vektormező: R[3] R[3] u(x, y, z) = xyi + (y − _z[2])j + (2z −_ _xy)k_ Határozza meg az u integrálját a) az origó középpontú, x-y síkban felvő egységkörvonalon a z tengely pozitív fele felől nézve pozitív körüljárási irányban b) ezen körvonal y ≥ 0 félkörén az előbbivel megegyező irányban. _Megoldás. A paraméterezés legyen r(t) = cos ti + sin tj, ekkor_ a) � � 2π u · dr = (cos t sin ti + sin tj − cos t sin tk) · (− sin ti + cos tj) dt 0 = � 2π �cos t sin t − cos t sin[2] _t�_ dt = 0 0 b) � � _π_ u · dr = 0 �cos t sin t − cos t sin[2] _t�_ dt = 0. 15. Integráljuk az u(x, y, z) = (2xy − _z)i + (x[2]_ + z)j + (y − _x)k vektormezőt az_ [1] 9 _[x][2][ +][ 1]16_ _[y][2][ = 1,]_ _z = 2 egyenletrendszerrel megadott ellipszisen a z tengely pozitív fele irányából nézve_ pozitív körüljárás szerint. _Megoldás. A deriváltmátrix_   _∂ux_ _∂ux_ _∂ux_ _∂x_ _∂y_ _∂z_ _∂uy_ _∂uy_ _∂uy_ _∂x_ _∂y_ _∂z_ _∂uz_ _∂uz_ _∂uz_ _∂x_ _∂y_ _∂z_   =  2y 2x _−1_  2x 0 1  _,_ _−1_ 1 0 ami szimmetrikus, tehát u potenciálos. A megadott görbe zárt, tehát a Newton-Leibniztétel alapján az integrál 0. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A négy szakaszon</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyaránt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (konstans).</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szögű forgásszimmetria miatt a két tengelyre vonatkozó tehetetlenségi nyomaték</span></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megegyezik. Pl. az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelyre nézve:</span></p> <p style="top:120.4pt;left:95.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Θ =</span></p> <p style="top:107.9pt;left:120.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:132.2pt;left:125.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:112.3pt;left:138.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i></p> <p style="top:128.6pt;left:137.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></p> <p style="top:112.3pt;left:152.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:128.6pt;left:154.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:107.9pt;left:206.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:132.2pt;left:212.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:112.3pt;left:225.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i></p> <p style="top:128.6pt;left:224.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></p> <p style="top:112.3pt;left:238.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:128.6pt;left:241.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:107.9pt;left:322.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:132.2pt;left:328.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:112.3pt;left:341.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i></p> <p style="top:128.6pt;left:340.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></p> <p style="top:112.3pt;left:355.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:128.6pt;left:357.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:107.9pt;left:427.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:132.2pt;left:433.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:112.3pt;left:446.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i></p> <p style="top:128.6pt;left:445.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></p> <p style="top:112.3pt;left:460.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:128.6pt;left:462.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:150.5pt;left:107.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ma</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:158.7pt;left:128.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> <p style="top:138.1pt;left:145.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:162.3pt;left:151.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:140.6pt;left:162.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:150.5pt;left:168.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></p> <p style="top:140.6pt;left:227.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:150.5pt;left:235.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ma</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:158.7pt;left:271.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> <p style="top:137.6pt;left:287.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:150.5pt;left:295.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:158.7pt;left:310.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></sup></p> <p style="top:158.7pt;left:348.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup></p> <p style="top:137.6pt;left:375.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:150.5pt;left:386.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ma</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:158.7pt;left:407.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:181.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> AB</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szakasz mentén,</span></p> <p style="top:195.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:214.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Használjuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterezést (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), ezzel</span></p> <p style="top:233.8pt;left:94.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:246.2pt;left:106.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:233.8pt;left:149.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:258.0pt;left:155.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ˙</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:274.8pt;left:137.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:262.4pt;left:149.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:286.6pt;left:155.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">((</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:303.4pt;left:137.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:291.0pt;left:149.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:315.2pt;left:155.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(12 + 14</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 19</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:334.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">14. Legyen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">→</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az alábbi vektormező:</span></p> <p style="top:360.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:386.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Határozza meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> integrálját</span></p> <p style="top:400.4pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) az origó középpontú,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban felvő egységkörvonalon a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely pozitív fele felől</span></p> <p style="top:414.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nézve pozitív körüljárási irányban</span></p> <p style="top:431.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) ezen körvonal</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> félkörén az előbbivel megegyező irányban.</span></p> <p style="top:450.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A paraméterezés legyen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ekkor</span></p> <p style="top:465.0pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span></p> <p style="top:484.1pt;left:128.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:496.6pt;left:140.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:484.1pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:508.3pt;left:190.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:496.6pt;left:204.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:525.2pt;left:172.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:512.7pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:536.9pt;left:190.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:515.2pt;left:206.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:525.2pt;left:212.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:515.2pt;left:321.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:525.2pt;left:328.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:555.2pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span></p> <p style="top:572.6pt;left:126.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:585.1pt;left:138.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:572.6pt;left:182.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:596.8pt;left:188.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:575.1pt;left:200.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:585.1pt;left:206.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:575.1pt;left:314.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:585.1pt;left:322.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:616.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15. Integráljuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:623.3pt;left:462.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:623.3pt;left:493.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:630.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletrendszerrel megadott ellipszisen a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely pozitív fele irányából nézve</span></p> <p style="top:644.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pozitív körüljárás szerint.</span></p> <p style="top:664.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A deriváltmátrix</span></p> <p style="top:681.5pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:699.0pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:687.7pt;left:114.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></p> <p style="top:696.6pt;left:116.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:687.7pt;left:141.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></p> <p style="top:696.6pt;left:143.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:687.7pt;left:168.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></p> <p style="top:696.6pt;left:170.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂z</span></i></p> <p style="top:703.7pt;left:114.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:713.5pt;left:116.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:703.7pt;left:141.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:713.5pt;left:143.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:703.7pt;left:168.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:713.5pt;left:170.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂z</span></i></p> <p style="top:720.6pt;left:114.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">z</span></i></p> <p style="top:729.4pt;left:116.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:720.6pt;left:141.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">z</span></i></p> <p style="top:729.4pt;left:143.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:720.6pt;left:168.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">z</span></i></p> <p style="top:729.4pt;left:170.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂z</span></i></p> <p style="top:681.5pt;left:183.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:699.0pt;left:183.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:684.5pt;left:206.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:702.0pt;left:206.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:691.9pt;left:214.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:691.9pt;left:237.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:691.9pt;left:260.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:706.3pt;left:214.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:706.3pt;left:241.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:706.3pt;left:265.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:720.7pt;left:212.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:720.7pt;left:241.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:720.7pt;left:265.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:684.5pt;left:275.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:702.0pt;left:275.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:748.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ami szimmetrikus, tehát</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> potenciálos. A megadott görbe zárt, tehát a Newton-Leibniz-</span></p> <p style="top:763.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tétel alapján az integrál</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_249.png	A négy szakaszon [(1)] egyaránt a (konstans). szögű forgásszimmetria miatt a két tengelyre vonatkozó. hetetlenségi nyomaték 9- [Zaa [ D8a-gad ; [ EÉCYaN ; [ DŐCI s gytade -EE [GR-a a TÉ [ú l9 22 13. Miaz ulr.y.2) — (y-42jíd (r 4-2)j4 (r -4-y)k vektormező integrálja az AB szakasz mentén, ha AZ (1.-2.3), B— (2.1.492 . Megoldás. Használjuk az r(t) — (1—2)4-3k) 4-t1--3)--k) paraméterezést (£ € [1. 1]), ezzel fra-ae- [ atetg)-se9ae ll sA LZF FŰ E 2F3Ű 14354 19dt - ffaz4149— 19. 14. Legyen u : RŐ — R" az alábbi vektormező: ú(r y.2) — 2vi 4 [y— N 4 (22 — ay)k a) az origó középpontú, 2-y síkban felvő egységkörvonalon a : tengely pozitív fele felől mézve pozítív körüljárási irányban 1) ezen körvonal y 2 0 félkörén az előbbivel megegyező irányban. .Megoldás. A paraméterezés legyen r(!) — costi 4. sintj, ekkor a) 2 fő (eostsint — costsitét) ar 2ry— 24 ( 4 2j 4 [y — 2)k vektormezőt az 322 4 9? Megoldás. A deriváltmátrix ami szímmetrikus, tehát u potenciálos. A megadott görbe zárt, tehát a Newton-Leibniz- tétel alapján az integrál 0.
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 3. feladatsor: Görbe ívhossza, görbementi integrál (megoldás) 1. Mi az r(t) = t3 3 i + 6 2 5 t 5 2j + 9 2t2k görbe ívhossza a t ∈[1, 2] intervallumon? 5 2j + 9 2 Megoldás. A megadott függvény diﬀerenciálható, tehát az ívhossz a derivált hosszának integrálásával áll elő: I = Z 2 Z 1 \|˙r(t)\| dt 1 Z 2 t2i + 3 √ 2t 3 2j + 9tk dt k dt 1 Z 2 √ t4 + 18t3 + 81t2 dt 1 Z 2 3 1 (t2 + 9t) dt = 23 −13 1 Z 2 3 1 (t2 + 9t) dt = 23 −13 + 9(22 −1) 2 = 95 6 . = 95 6 2. Tekintsük a síkon az r(t) = f(t) cos ti + f(t) sin tj paraméteres egyenletű spirált (csigavonal), ahol f diﬀerenciálható függvény. Mekkora a görbe 0 ≤t ≤2π szakaszának (azaz egy körülfordulásnak) ívhossza, ha a) f(t) = t (arkhimédeszi spirál) b) f(t) = αt valamilyen rögzített α > 0 értékkel (logaritmikus spirál) Megoldás. Tetszőleges diﬀerenciálható f mellett az ívhossz így számolható: Z 2π 0 Z 2π 0 Z 2π I = \|˙r(t)\| dt \|(f ′(t) cos t −f(t) sin t) i + (f ′(t) sin t + f(t) cos t) j\| dt q f ′(t)2 + f(t)2 dt. a) Ha f(t) = t, akkor f ′(t) = 1, tehát Z 2π √ 0 "t √ 1 + t2 dt √ 1 + t2 + arsh t I = #2π = π √ 1 + 4π2 + 1 2 arsh 2π √ 1 + 4π2 + 1 2 b) Ha f(t) = αt, akkor f ′(t) = (log α)αt, tehát I = Z 2π q 1 + log2 ααt dt 2π q 1 + log2 α log α αt q 1 + log2 α q 1 + log2 α log α (α2π −1) q 1 + log2 α	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 3. feladatsor: Görbe ívhossza, görbementi integrál (megoldás) _√_ 1. Mi az r(t) = _[t][3]_ 2 52 j + [9] 3 [i][ +][ 6] 5 _[t]_ 2 _[t][2][k][ görbe ívhossza a][ t][ ∈]_ [[1][,][ 2] intervallumon?] _Megoldás. A megadott függvény differenciálható, tehát az ívhossz a derivált hosszának_ integrálásával áll elő: � 2 _I =_ 1 _[\|][r][˙][(][t][)][\|][ d][t]_ = � 2 _t2i + 3√2t_ 32 j + 9tk dt 1 �� 2 _√_ � = _t[4]_ + 18t[3] + 81t[2] dt 1 � 2 = + [9(2][2][ −] [1)] = [95] 1 [(][t][2][ + 9][t][) d][t][ = 2][3][ −]3 [1][3] 2 6 _[.]_ 2. Tekintsük a síkon az r(t) = f (t) cos ti + f (t) sin tj paraméteres egyenletű spirált (csigavonal), ahol f differenciálható függvény. Mekkora a görbe 0 ≤ _t ≤_ 2π szakaszának (azaz egy körülfordulásnak) ívhossza, ha a) f (t) = t (arkhimédeszi spirál) b) f (t) = α[t] valamilyen rögzített α > 0 értékkel (logaritmikus spirál) _Megoldás. Tetszőleges differenciálható f mellett az ívhossz így számolható:_ � 2π _I =_ _\|r˙(t)\| dt_ 0 � 2π = _\|(f_ _[′](t) cos t −_ _f_ (t) sin t) i + (f _[′](t) sin t + f_ (t) cos t) j\| dt 0 � 2π � = _f_ _[′](t)[2]_ + f (t)[2] dt. 0 a) Ha f (t) = t, akkor f _[′](t) = 1, tehát_ 1 + t[2] dt � 2π _I =_ 0 _√_ � _t_ = _√_ 2π 1 + t[2] + arsh t � 2 0 1 + 4π[2] + [1] 2 [arsh 2][π] _√_ = π b) Ha f (t) = α[t], akkor f _[′](t) = (log α)α[t], tehát_ � 2π _I =_ 0 � 1 + log[2] _αα[t]_ dt 2π  =  0 � 1 + log[2] _α_ (α[2][π] _−_ 1) log α � 1 + log[2] _α_ _α[t]_ log α =   -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">3. feladatsor: Görbe ívhossza, görbementi integrál</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">(megoldás)</span></b></p> <p style="top:150.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup></p> <p style="top:158.2pt;left:147.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 6</span></sup></p> <p style="top:142.8pt;left:177.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:149.4pt;left:184.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:158.2pt;left:179.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:148.0pt;left:195.7pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">5</span></p> <p style="top:154.4pt;left:195.7pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span></sup></p> <p style="top:158.2pt;left:220.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe ívhossza a</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> intervallumon?</span></sup></p> <p style="top:170.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megadott függvény diﬀerenciálható, tehát az ívhossz a derivált hosszának</span></p> <p style="top:184.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">integrálásával áll elő:</span></p> <p style="top:215.1pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:202.7pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:226.9pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:243.7pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:231.3pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:255.5pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:233.4pt;left:147.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></p> <p style="top:233.4pt;left:183.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:243.7pt;left:193.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:240.3pt;left:204.5pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></p> <p style="top:246.7pt;left:204.5pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:233.4pt;left:245.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:272.3pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:259.9pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:284.1pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:261.9pt;left:147.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:272.3pt;left:156.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 18</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 81</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:302.5pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:290.0pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:314.2pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:310.7pt;left:232.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:302.5pt;left:256.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9(2</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup></p> <p style="top:310.7pt;left:289.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:302.5pt;left:320.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">95</span></sup></p> <p style="top:310.7pt;left:336.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:331.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Tekintsük a síkon az</span></p> <p style="top:357.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:383.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paraméteres egyenletű spirált (csigavonal), ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható függvény. Mekkora a</span></p> <p style="top:397.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbe</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szakaszának (azaz egy körülfordulásnak) ívhossza, ha</span></p> <p style="top:411.9pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (arkhimédeszi spirál)</span></p> <p style="top:428.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">valamilyen rögzített</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékkel (logaritmikus spirál)</span></p> <p style="top:447.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Tetszőleges diﬀerenciálható</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mellett az ívhossz így számolható:</span></p> <p style="top:478.1pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:465.6pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:489.9pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:478.1pt;left:152.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:506.7pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:494.3pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:518.5pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:506.7pt;left:152.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:535.3pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:522.9pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:547.1pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:524.3pt;left:152.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:535.3pt;left:162.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t.</span></i></p> <p style="top:565.1pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:596.4pt;left:128.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:584.0pt;left:150.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:608.2pt;left:156.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:586.0pt;left:172.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:596.4pt;left:182.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:629.9pt;left:138.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:613.9pt;left:150.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:611.9pt;left:162.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:621.8pt;left:172.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ arsh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></p> <p style="top:638.1pt;left:197.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:613.9pt;left:244.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:645.5pt;left:250.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:661.9pt;left:138.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π</span></i></p> <p style="top:651.5pt;left:157.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:661.9pt;left:167.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:670.1pt;left:221.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arsh 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup></p> <p style="top:691.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (log</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:722.9pt;left:128.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:710.5pt;left:150.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:734.7pt;left:156.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:710.9pt;left:172.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:722.9pt;left:182.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + log</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αα</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:760.2pt;left:138.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:741.3pt;left:150.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:759.2pt;left:150.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:739.9pt;left:158.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:751.3pt;left:168.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + log</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:768.4pt;left:176.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">log</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i></p> <p style="top:760.2pt;left:219.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup></p> <p style="top:741.3pt;left:230.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:759.2pt;left:230.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:743.8pt;left:237.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:778.8pt;left:237.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:760.2pt;left:250.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:739.9pt;left:263.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:751.3pt;left:273.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + log</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:768.4pt;left:281.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">log</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i></p> <p style="top:760.2pt;left:324.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> </div>	page_250.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 3. feladatsor: Görbe ívhossza, görbementi integrál (megoldás) 1. Miaze() — §14 2É 2 k gőrbe ívhossza a ! € [1.2 intervallumon? .Megoldás. A megadott fű dittes 1- főttoras - lér sv8n VIKGEZSSEETÉ ] te 22- , 921 95 - é9 2L, XZV B meiálható, tehát az ívhossz a derívált hosszának 5 90ját 2. Tekintsük a síkon az el0) — fl)eosti 4 f(ősintj görbe 0 £ t £ a) /(0 b) 79 . Megoldás. Tetszöleges dilferenciálható / me tű spirált (csisavonal), ahol / differenciálható függvény. . Mekkora a. szakaszának (azaz egy körülfordulásnak) ívhossza, ha 1 (arkhimédeszi spírál) a valamilyen rögzített a — 0 éri kel (logaritmikus spirál) ett az frhossz így számolható: 1- Á"xr(mm - [$lr esst — fősingi4 ( sint 4 J cos031dt a) Ha f(4) — t, akkor fít) — 1, tehát 1- [7 vIvEa p m_m]" 2 1) Ha f(4) — at, akkor f1(t) — (logaja, tehát e 1 loszaat dt . [F Floga , loga
3. Integráljuk az f(x, y, z) = √1 + 4x + 9yz skalármezőt az r(t) = t2i+tj+t3k görbe mentén t = 0 és t = 1 paraméterértékek között. Megoldás. A skalármező folytonos, a paraméterezés diﬀerenciálható, tehát az alábbi (egyváltozós) integrállal számíthatjuk a görbementi integrált: f ds = Z 1 Z 0 f(r(t))\|˙r(t)\| dt 0 Z 1 √ 1 + 4t2 + 9t4 2ti + j + 3t2k dt 2ti + j + 3t2k 0 Z 1 Z 1 3 0 (1 + 4t2 + 9t4) dt = 1 + 4 4 3 + 9 5 9 5 = 62 15 62 15. 4. Egy vékony L hosszúságú drót vonalmenti sűrűsége egyik végétől a másikig lineárisan vál- tozik µ és 2µ között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközéppontja? Megoldás. Válasszunk úgy koordinátarendszert, hogy a kör középpontja az origó, és a drót két (egybeeső) vége az x tengely pozitív felén van. Ekkor egy paraméterezés r(t) = r cos ti+ r sin t, t ∈[0, 2π], ahol r = L 2π, a sűrűség µ(1 + t 2π). A tömegközéppont koordinátáit az elsőrendű nyomatékok és a tömeg hányadosa adják. \|˙r(t)\| = r\| −sin ti + cos tj\| = r alapján Z 2π 1 + t 2π r cos tµ 1 + t 2π r dt Z 2π µ 1 + t r dt r cos tµ 2π xtk = 0 3πrµ = 0 1 + t 2 2π r dt és Z 2π 1 + t 2π r sin tµ 1 + t 2π r dt Z 2π µ 1 + t r dt r sin tµ ytk = 2π = −r2µ 3πrµ −r2µ 3πrµ = −r 3π r 3π. 1 + t 2 2π r dt 5. Mennyi az u(x, y, z) = (y2 −x2)i + 2yzj −x2k vektormező integrálja az r(t) = ti + t2j + t3k görbe mentén t = 0-tól t = 1-ig? Megoldás. Folytonos vektormezőt kell integrálni diﬀerenciálható paraméterezéssel megadott görbén, ezt egyváltozós integrállal lehet számolni: u · dr = Z 1 Z 0 u(r) · ˙r(t) dt 0 Z 1 Z 1 0 ((t4 −t2)i + 2t5j −t2k) · (i + 2tj + 3t2k) dt 0 Z 1 Z 1 3 0 (−t2 −2t5 + 4t6) dt = −1 1 3 −2 · 1 5 1 5 + 4 · 1 7 1 7 = −17 105 17 105. 6. Integráljuk az u : R3 →R3 u(x, y, z) = yi + xj + 2zk vektormezőt az r : R →R3 r(t) = 2t+1 −1 i + t 4 + 2t2j + sin π 4 t e(1−t)2k térgörbe mentén a t = 0 és t = 1 paraméterértékeknek megfelelő pontok között.	3. Integráljuk az f (x, y, z) = _[√]1 + 4x + 9yz skalármezőt az r(t) = t[2]i_ + _tj_ + _t[3]k görbe mentén_ _t = 0 és t = 1 paraméterértékek között._ _Megoldás. A skalármező folytonos, a paraméterezés differenciálható, tehát az alábbi (egy-_ változós) integrállal számíthatjuk a görbementi integrált: � � 1 _f ds =_ 0 _[f]_ [(][r][(][t][))][\|][r][˙][(][t][)][\|][ d][t] 1 _√_ = � 1 + 4t[2] + 9t[4] 2ti + j + 3t2k dt 0 �� 1 = 0 [(1 + 4][t][2][ + 9][t][4][) d][t][ = 1 + 4]3 [+ 9]5 [= 62]15[.] 4. Egy vékony L hosszúságú drót vonalmenti sűrűsége egyik végétől a másikig lineárisan változik µ és 2µ között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközéppontja? _Megoldás. Válasszunk úgy koordinátarendszert, hogy a kör középpontja az origó, és a drót_ két (egybeeső) vége az x tengely pozitív felén van. Ekkor egy paraméterezés r(t) = r cos ti+ _r sin t, t ∈_ [0, 2π], ahol r = _L_ _t_ 2π [, a sűrűség][ µ][(1 +] 2π [). A tömegközéppont koordinátáit az] elsőrendű nyomatékok és a tömeg hányadosa adják. \|r˙(t)\| = r\| − sin ti + cos tj\| = r alapján � 2π � � _r cos tµ_ 1 + _[t]_ _r dt_ 0 2π 0 � 2π � � = 3πrµ [= 0] _µ_ 1 + _[t]_ _r dt_ 0 2π � 2π � � _r sin tµ_ 1 + _[t]_ _r dt_ 0 2π � 2π � � = _[−]3πrµ[r][2][µ]_ [=][ −]3[r]π _[.]_ _µ_ 1 + _[t]_ _r dt_ 0 2π és _xtk =_ _ytk =_ 5. Mennyi az u(x, y, z) = (y[2] _−_ _x[2])i + 2yzj −_ _x[2]k vektormező integrálja az r(t) = ti + t[2]j + t[3]k_ görbe mentén t = 0-tól t = 1-ig? _Megoldás. Folytonos vektormezőt kell integrálni differenciálható paraméterezéssel megadott_ görbén, ezt egyváltozós integrállal lehet számolni: � � 1 u · dr = 0 [u][(][r][)][ ·][ ˙][r][(][t][) d][t] � 1 = 0 [((][t][4][ −] _[t][2][)][i][ + 2][t][5][j][ −]_ _[t][2][k][)][ ·][ (][i][ + 2][t][j][ + 3][t][2][k][) d][t]_ � 1 = 0 [(][−][t][2][ −] [2][t][5][ + 4][t][6][) d][t][ =][ −][1]3 _[−]_ [2][ ·][ 1]5 [+ 4][ ·][ 1]7 [=][ −]105[17] _[.]_ 6. Integráljuk az u : _→_ R[3] R[3] u(x, y, z) = yi + xj + 2zk vektormezőt az r : _→_ R R[3] r(t) = �2[t][+1] _−_ 1� i + _t_ � _π_ � _e[(1][−][t][)][2]k_ 4 + 2t[2] [j][ + sin] 4 _[t]_ térgörbe mentén a t = 0 és t = 1 paraméterértékeknek megfelelő pontok között. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Integráljuk az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> skalármezőt az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe mentén</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterértékek között.</span></p> <p style="top:93.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A skalármező folytonos, a paraméterezés diﬀerenciálható, tehát az alábbi (egy-</span></p> <p style="top:107.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">változós) integrállal számíthatjuk a görbementi integrált:</span></p> <p style="top:126.9pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:139.4pt;left:120.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">s</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:126.9pt;left:157.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:151.1pt;left:162.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:168.0pt;left:144.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:155.5pt;left:157.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:179.8pt;left:162.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:157.6pt;left:173.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:168.0pt;left:183.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:157.6pt;left:250.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:157.6pt;left:322.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:196.6pt;left:144.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:184.1pt;left:157.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:208.4pt;left:162.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1 + 4</span></sup></p> <p style="top:204.8pt;left:295.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 9</span></sup></p> <p style="top:204.8pt;left:317.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 62</span></sup></p> <p style="top:204.8pt;left:341.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:226.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Egy vékony</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> L</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> hosszúságú drót vonalmenti sűrűsége egyik végétől a másikig lineárisan vál-</span></p> <p style="top:240.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tozik</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközép-</span></p> <p style="top:255.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pontja?</span></p> <p style="top:274.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Válasszunk úgy koordinátarendszert, hogy a kör középpontja az origó, és a drót</span></p> <p style="top:289.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">két (egybeeső) vége az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely pozitív felén van. Ekkor egy paraméterezés</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:303.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:302.0pt;left:227.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">L</span></i></p> <p style="top:310.9pt;left:226.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a sűrűség</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span></sup></p> <p style="top:302.0pt;left:335.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></p> <p style="top:310.9pt;left:332.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tömegközéppont koordinátáit az</span></sup></p> <p style="top:318.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elsőrendű nyomatékok és a tömeg hányadosa adják.</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\| −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alapján</span></p> <p style="top:363.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:337.5pt;left:138.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:361.8pt;left:143.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:350.0pt;left:160.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> tµ</span></i></p> <p style="top:337.0pt;left:198.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:350.0pt;left:206.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:358.2pt;left:227.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:337.0pt;left:241.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:350.0pt;left:250.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:365.3pt;left:153.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:389.6pt;left:158.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:377.8pt;left:174.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i></p> <p style="top:364.8pt;left:183.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:377.8pt;left:191.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:386.0pt;left:212.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:364.8pt;left:226.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:377.8pt;left:236.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:363.3pt;left:273.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:355.3pt;left:297.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:371.5pt;left:287.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πrµ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></sup></p> <p style="top:407.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></p> <p style="top:450.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:424.5pt;left:137.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:448.7pt;left:142.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:436.9pt;left:159.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> tµ</span></i></p> <p style="top:424.0pt;left:196.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:436.9pt;left:203.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:445.1pt;left:225.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:424.0pt;left:239.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:436.9pt;left:248.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:452.3pt;left:151.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:476.5pt;left:157.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:464.7pt;left:173.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i></p> <p style="top:451.8pt;left:182.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:464.7pt;left:189.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:472.9pt;left:211.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:451.8pt;left:225.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:464.7pt;left:234.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:450.3pt;left:271.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i></sup></p> <p style="top:458.5pt;left:285.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πrµ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup></p> <p style="top:458.5pt;left:339.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:496.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:510.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbe mentén</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-tól</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ig?</span></p> <p style="top:530.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Folytonos vektormezőt kell integrálni diﬀerenciálható paraméterezéssel megadott</span></p> <p style="top:544.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbén, ezt egyváltozós integrállal lehet számolni:</span></p> <p style="top:563.3pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:575.8pt;left:120.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:563.3pt;left:164.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:587.5pt;left:169.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ˙</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:604.4pt;left:151.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:591.9pt;left:164.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:616.1pt;left:169.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">((</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:633.0pt;left:151.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:620.5pt;left:164.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:644.7pt;left:169.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:641.2pt;left:304.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></sup></p> <p style="top:641.2pt;left:341.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></sup></p> <p style="top:641.2pt;left:378.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">17</span></sup></p> <p style="top:641.2pt;left:411.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">105</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:664.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Integráljuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">→</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:690.4pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:716.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektormezőt az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:745.2pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:735.2pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.2pt;left:147.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:735.2pt;left:187.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.2pt;left:195.6pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:737.1pt;left:230.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:753.4pt;left:215.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span></sup></p> <p style="top:732.2pt;left:286.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:753.4pt;left:295.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:732.2pt;left:307.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.2pt;left:316.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(1</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:775.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">térgörbe mentén a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterértékeknek megfelelő pontok között.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_251.png	6. Integráljuk az f(r, y,2) — VTT ET 77 skalármezőt az r() — Éi--tj4-Bk görbe menté 0 és £ — 1 paraméterértékek között. . Megoldás. A skalármező folytonos, a paraméterezés díflerenciálható, tehát az alábbi (egy- változós) integrállal számíthatjuk a görbementi intesrált: J1as- [; retoyetolat - [[E TRa 434 sék[at tozik ja és 21 között. A drótból körvonalat hajlítunk. Hol lesz a kapott test tömegközép- rsint, ! € [0,2a]. ahol r — £. a sűrűség a(14.£). A tömegközéppont koordinátáit az M. görbe mentén £ a2J14-2y2j — 22k vektormező integrálja az r() — 1i4-Éj4-Ék 1 . Megoldás. Folytonos vektorn görbén, ezt egyváltozás integrállal lehet számolni: a [/909-K94. - [/d6- é- 44294 38194£ MR 2644 1 f/eg-atsaba to lá4 szőt kell integrálni dfferenciálható paraméterezéssel megadott Integráljuk az u : R? — R ulry.2) á j 22k vektormezőt az r : R R r -R C" t ss (? térgörbe mentén a paraméterértékeknek megfelelő pontok között. 2
Megoldás. ∂2z ∂y ∂x rot u(x, y, z) = ∂2z ∂y −∂x ∂z ∂z i + ∂y ∂z −∂2z ∂x ∂x j + ∂x ∂x −∂y ∂y ∂y k = 0, tehát létezik potenciálfüggvény. Valóban, f(x, y, z) = xy + z2 választással u = grad f. Használhatjuk a Newton-Leibniz-tételt, ekkor az integrál meghatározásához elég a két végpontot ismerni: r(0) = i és r(1) = 3i + 1 6j + 1 √ 2k, tehát 1 6j + 1 √ 2k, tehát Z u · dr = f(r(1)) −f(r(0)) = 1. 7. Mi az u(x, y) = −y x2+y2i + x x2+y2j vektormező integrálja a) az origó körüli R sugarú kör mentén pozitív irányítással b) az origó körüli R sugarú kör mentén negatív irányítással c) az (5, 9) pont körüli 2 sugarú kör mentén pozitív irányítással d) az r(t) = αi + tj egyenes mentén (α > 0) t = −∞-től t = +∞-ig? Megoldás. a) Egy diﬀerenciálható paraméterezés r(t) = R cos ti+R sin tj, ahol t ∈[0, 2π], a paraméter növekedése pozitív irányú körülfordulást eredményez. Az integrál Z 2π 0 Z 2π −R sin t u · dr = R sin t R2 i + R cos t R2 R2 · (−R sin ti + R cos tj) dt 1 dt = 2π. b) Az előbbi görbe megfordításán integráljuk ugyanazt a vektormezőt, tehát az integrál −2π. c) ∂x x x2 + y2 −∂ ∂y x x2 + y2 −∂ ∂y −y x2 + y2 = (x2 + y2) −(x · 2x) (x2 + y2)2 + y2) −(x · 2x) (x2 + y2)2 + (x2 + y2) −(y · 2y) (x2 + y2)2 + y ) (y 2y) (x2 + y2)2 = 0 ha (x, y) ̸= (0, 0). A megadott körvonal körül találunk olyan négyzetet, ami nem tartalmazza az origót, ezen létezik potenciál. A körvonal zárt görbe, tehát a NewtonLeibniz-tétel szerint az integrál 0. d) A görbe végtelen hosszú, de diﬀerenciálható függvénnyel adott (tehát lokálisan rektiﬁ- kálható), így a vonalmenti integrál improprius integrálként értelmes lehet: Z ∞ u · dr = −t α2 + t2i + α α2 + t2j · j dt −∞ Z ∞ −∞ 1 + 2 dt = arctan t t=∞ t=−∞= π. További gyakorló feladatok 8. Mi az r(t) = et cos ti + et sin tj + etk görbe ívhossza a t ∈[0, 2] intervallumon? Megoldás. Z 2 I = Z 0 \|˙r(t)\| dt 0 Z 2 et cos t −et sin t et sin t + et cos t i + j + etk dt k dt 0 Z 2 √ 3et = √ 3(e2 −1).	_Megoldás._ rot u(x, y, z) = �∂2z _∂y_ _[−]_ _[∂x]∂z_ � i + �∂y _∂z_ _[−]_ _[∂]∂x[2][z]_ � j + �∂x _∂x_ _[−]_ _[∂y]∂y_ � k = 0, tehát létezik potenciálfüggvény. Valóban, f (x, y, z) = xy + z[2] választással u = grad f . Használhatjuk a Newton-Leibniz-tételt, ekkor az integrál meghatározásához elég a két végpontot ismerni: r(0) = i és r(1) = 3i + [1] _√1_ 6 [j][ +] 2 [k][, tehát] � u · dr = f (r(1)) − _f_ (r(0)) = 1. 7. Mi az u(x, y) = _−y_ _x_ _x[2]+y[2]_ [i][ +] _x[2]+y[2]_ [j][ vektormező integrálja] a) az origó körüli R sugarú kör mentén pozitív irányítással b) az origó körüli R sugarú kör mentén negatív irányítással c) az (5, 9) pont körüli 2 sugarú kör mentén pozitív irányítással d) az r(t) = αi + tj egyenes mentén (α > 0) t = −∞-től t = +∞-ig? _Megoldás._ a) Egy differenciálható paraméterezés r(t) = R cos ti+R sin tj, ahol t ∈ [0, 2π], a paraméter növekedése pozitív irányú körülfordulást eredményez. Az integrál � � 2π � _−R sin t_ � u · dr = i + _[R][ cos][ t]j_ _· (−R sin ti + R cos tj) dt_ 0 _R[2]_ _R[2]_ � 2π = 1 dt = 2π. 0 b) Az előbbi görbe megfordításán integráljuk ugyanazt a vektormezőt, tehát az integrál _−2π._ c) _∂_ _x_ _−y_ + [(][x][2][ +][ y][2][)][ −] [(][y][ ·][ 2][y][)] = 0 _∂x_ _x[2]_ + y[2][ −] _∂y[∂]_ _x[2]_ + y[2][ = (][x][2][ +]([ y]x[2][2][)]+[ −] y[2][(])[x][2][ ·][ 2][x][)] (x[2] + y[2])[2] ha (x, y) ̸= (0, 0). A megadott körvonal körül találunk olyan négyzetet, ami nem tartalmazza az origót, ezen létezik potenciál. A körvonal zárt görbe, tehát a NewtonLeibniz-tétel szerint az integrál 0. d) A görbe végtelen hosszú, de differenciálható függvénnyel adott (tehát lokálisan rektifikálható), így a vonalmenti integrál improprius integrálként értelmes lehet: � � _∞_ � _−t_ _α_ � u · dr = _· j dt_ _−∞_ _α[2]_ + t[2] [i][ +] _α[2]_ + t[2] [j] � _∞_ 1 1 � �t=∞ = 2 dt = arctan _[t]_ _−∞_ _α_ 1 + � _t_ � _α_ _t=−∞_ [=][ π.] _α_ ## További gyakorló feladatok 8. Mi az r(t) = e[t] cos ti + e[t] sin tj + e[t]k görbe ívhossza a t ∈ [0, 2] intervallumon? _Megoldás._ � 2 _I =_ 0 _[\|][r][˙][(][t][)][\|][ d][t]_ = � 2 �e[t] cos t − _e[t]_ sin t� i + �e[t] sin t + e[t] cos t� j + e[t]k dt 0 �� 2 _√_ _√_ � = 3e[t] = 3(e[2] _−_ 1). 0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:87.6pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:71.6pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:95.8pt;left:196.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></sup></p> <p style="top:95.8pt;left:230.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂z</span></i></p> <p style="top:71.6pt;left:244.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:87.6pt;left:254.3pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:71.6pt;left:272.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:95.8pt;left:281.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup></p> <p style="top:95.8pt;left:314.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:71.6pt;left:331.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:87.6pt;left:341.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:71.6pt;left:359.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:95.8pt;left:368.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></sup></p> <p style="top:95.8pt;left:399.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:71.6pt;left:413.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:87.6pt;left:423.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:117.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát létezik potenciálfüggvény. Valóban,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">választással</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = grad</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:131.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Használhatjuk a Newton-Leibniz-tételt, ekkor az integrál meghatározásához elég a két vég-</span></p> <p style="top:146.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pontot ismerni:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:153.4pt;left:279.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:144.6pt;left:308.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:147.6pt;left:304.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:154.3pt;left:311.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></sup></p> <p style="top:161.9pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:174.4pt;left:118.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1))</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)) = 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:200.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:198.7pt;left:168.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:207.9pt;left:161.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:199.1pt;left:216.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:207.9pt;left:206.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja</span></sup></p> <p style="top:216.3pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) az origó körüli</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör mentén pozitív irányítással</span></p> <p style="top:232.7pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) az origó körüli</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör mentén negatív irányítással</span></p> <p style="top:249.1pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 9)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pont körüli</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör mentén pozitív irányítással</span></p> <p style="top:265.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenes mentén (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = +</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ig?</span></p> <p style="top:283.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:297.4pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) Egy diﬀerenciálható paraméterezés</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a paraméter</span></p> <p style="top:311.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">növekedése pozitív irányú körülfordulást eredményez. Az integrál</span></p> <p style="top:326.2pt;left:128.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:338.7pt;left:140.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:326.2pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:350.4pt;left:190.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:325.7pt;left:206.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></p> <p style="top:346.9pt;left:228.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:338.7pt;left:257.3pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup></p> <p style="top:346.9pt;left:286.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:338.7pt;left:310.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:325.7pt;left:314.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:338.7pt;left:324.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:367.7pt;left:172.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:355.2pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:379.4pt;left:190.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:367.7pt;left:206.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π.</span></i></p> <p style="top:393.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) Az előbbi görbe megfordításán integráljuk ugyanazt a vektormezőt, tehát az integrál</span></p> <p style="top:407.9pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:424.3pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span></p> <p style="top:445.0pt;left:131.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:461.3pt;left:128.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:445.0pt;left:159.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:461.3pt;left:144.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup></p> <p style="top:461.3pt;left:197.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:445.0pt;left:223.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:461.3pt;left:213.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:461.3pt;left:291.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:453.1pt;left:369.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:461.3pt;left:404.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:453.1pt;left:482.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></p> <p style="top:481.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:481.5pt;left:204.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A megadott körvonal körül találunk olyan négyzetet, ami nem</span></p> <p style="top:495.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tartalmazza az origót, ezen létezik potenciál. A körvonal zárt görbe, tehát a Newton-</span></p> <p style="top:510.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Leibniz-tétel szerint az integrál</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:526.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) A görbe végtelen hosszú, de diﬀerenciálható függvénnyel adott (tehát lokálisan rektiﬁ-</span></p> <p style="top:541.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kálható), így a vonalmenti integrál improprius integrálként értelmes lehet:</span></p> <p style="top:555.5pt;left:128.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:567.9pt;left:140.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:555.5pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ∞</span></i></p> <p style="top:579.7pt;left:190.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−∞</span></i></p> <p style="top:555.0pt;left:207.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:559.9pt;left:227.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:576.1pt;left:216.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:559.9pt;left:286.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:576.1pt;left:272.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup></p> <p style="top:555.0pt;left:313.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:567.9pt;left:323.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:598.5pt;left:172.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:586.1pt;left:184.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ∞</span></i></p> <p style="top:610.3pt;left:190.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−∞</span></i></p> <p style="top:590.4pt;left:209.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:606.7pt;left:209.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:590.4pt;left:238.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:610.9pt;left:218.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></p> <p style="top:600.9pt;left:239.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:609.3pt;left:247.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></p> <p style="top:618.2pt;left:246.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></p> <p style="top:600.9pt;left:253.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:585.6pt;left:293.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:598.5pt;left:298.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arctan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:606.7pt;left:334.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:585.6pt;left:343.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:611.2pt;left:348.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−∞</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π.</span></i></sup></p> <p style="top:641.8pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:665.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe ívhossza a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> intervallumon?</span></p> <p style="top:683.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:709.8pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:697.4pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:721.6pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:738.4pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:726.0pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:750.2pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:728.1pt;left:147.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:728.5pt;left:150.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:738.4pt;left:156.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></p> <p style="top:728.5pt;left:235.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:738.4pt;left:243.3pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:728.5pt;left:261.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:738.4pt;left:267.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></p> <p style="top:728.5pt;left:346.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:738.4pt;left:354.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:728.1pt;left:389.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:767.1pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:754.6pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:778.8pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:756.7pt;left:147.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:767.1pt;left:156.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:756.7pt;left:187.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:767.1pt;left:197.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_252.png	. Megoldás. rotüln )— át létezik potenciálfüggvény. . Valóban, /(r.y.2) Használhatjuk a Newton-Leibniz-tételt, ekkor az integ pontot ismerni: r(0) — i és r(1) — 31 4 4 3k. tehát a0 f0e9)- 109 1. 7. Mi az ulr, y) — szat F szőz) vektormező integrálja a) az origó körüli A sugarú kör mentén pozitív ír 1) az origó körüli A sugarú kör m tással én negatív irányítással €) az (5.9) pont körüli 2 sugarú kör mentén pozitív irányítással d) az r(t) — ai 4- tj egyenes mentén (a — 0) oc-től . Megoldás. a) Egy diferenciálható paraméterezés r(! mövekedése pozitív irányú Reosti4-Rsin tj. ahol ! € [0, 22]. a paraméter körülfordulást eredményez. Az integrál fea (EE EE ) (-Rsinti 4 Reostjyát 2 [/ 142x. 1) Az előbbi görbe megfordításán integráljuk ugyanazt a vektormezőt, tehát az integrál 9 2 : 0 - [-E m t- 2) EE TTSET - RKZE (GE 0 Leibniz-tétel szerint az integrl 0. d) A gőrbe végtelen hosszú, de differenciálható függvénnyel adott (tehát lokálisan rektifi- kálható), így a vonalmenti integrál improprius integrálkó £ 4 rtelmes lehet: "További gyakorló feladatok 8. Mi az r(t) — €costi 4. e sintj 4 €"k görbe ívhossza a ! € [0,2) intervallumon? . Megoldás. FALOTT - f/lcsst - ésnis (Éánt4 é osja főv9 - v8é -n
√ 9. Mi az r(t) = (sinh t + cosh t)i + (cosh t −sinh t)j + 2tk görbe ívhossza a t ∈[0, ln 2] intervallumon? Megoldás. I = Z 2 Z 0 \|˙r(t)\| dt 0 Z 2 (cosh t + sinh t) i + (sinh t −cosh t) j + √ 2k dt k dt 0 Z 2 Z 2 2 0 2 cosh t = 3 3 2. 10. Mennyi az r(t) = 2 sin(t)i + 2 cos(t)j + t2 ! t 2 −ln t térgörbe 1 ≤t ≤√e paraméterértékeknek megfelelő részének ívhossza? Megoldás. Z √e 1 Z √e 1 Z √e I = \|˙r(t)\| dt t −1 t t + 1 t 2 cos ti −2 sin tj + dt Z √e "t2 #√e t s 2 + 1 t2 + t2 = s 2 + 1 t2 # t 2 + ln t = e 2. = e 2 11. Egy homogén tömegeloszlású vékony kötelet két végénél fogva lógatunk. Az általa meg- határozott görbe egy paraméterezése r(t) = ti + cosh tj, t ∈[−1, 1]. Hol van a kötél tömegközéppontja? Megoldás. A görbe az y tengelyre szimmetrikus, tehát xtk = 0. A másik komponens az elsőrendű nyomaték és a tömeg hányadosa: Z 1 ytk = Z −1 cosh t\|i + sinh tj\| dt 1Z 1 Z −1 \|i + sinh tj\| dt Z 1 Z 1 −1 cosh2 t dt Z 1 cosh t dt Z −1 cosh t dt = 1 + cosh 1 sinh 1 2 sinh 1 ≈1,197. = 1 + cosh 1 sinh 1 2 sinh 1 12. Homogén tömegeloszlású m tömegű vékony drótból a oldalú négyzet alakú keretet hajlítunk. Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatékát. Megoldás. Legyen a koordinátarendszer olyan, hogy a két átló az x illetve y tengelyre esik. Ekkor a négy oldal egy-egy paraméterezése (mindegyiknél t ∈[0, 1]): r(t) = a √ 2 a √ 2((1 −t)i + tj) r(t) = a √ 2 a √ 2(−ti + (1 −t)j) r(t) = a √ 2 a √ 2((−1 + t)i −tj) r(t) = a √ 2 a √ 2(ti + (−1 + t)j)	_√_ 9. Mi az r(t) = (sinh t + cosh t)i + (cosh t − sinh t)j + 2tk görbe ívhossza a t ∈ [0, ln 2] intervallumon? _Megoldás._ � 2 _I =_ 0 _[\|][r][˙][(][t][)][\|][ d][t]_ 2 _√_ � = (cosh t + sinh t) i + (sinh t − cosh t) j + 0 �� 2 = 0 [2 cosh][ t][ = 3]2[.] 2k dt �� 10. Mennyi az �t2 � r(t) = 2 sin(t)i + 2 cos(t)j + 2 _[−]_ [ln][ t] k térgörbe 1 ≤ _t ≤_ _[√]e paraméterértékeknek megfelelő részének ívhossza?_ _Megoldás._ � _[√]e_ _I =_ _\|r˙(t)\| dt_ 1 � _[√]e_ � � = 2 cos ti − 2 sin tj + _t −_ [1] k dt 1 �� _t_ �� _[√]e_ � � _[√]e_ � � � _t2_ �[√]e = 2 + [1] _t + [1]_ = = _[e]_ 1 _t[2][ +][ t][2][ =]_ 1 _t_ 2 [+ ln][ t] 2[.] 1 11. Egy homogén tömegeloszlású vékony kötelet két végénél fogva lógatunk. Az általa meghatározott görbe egy paraméterezése r(t) = ti + cosh tj, t ∈ [−1, 1]. Hol van a kötél tömegközéppontja? _Megoldás. A görbe az y tengelyre szimmetrikus, tehát xtk = 0. A másik komponens az_ elsőrendű nyomaték és a tömeg hányadosa: � 1 _−1_ [cosh][ t][\|][i][ + sinh][ t][j][\|][ d][t] _ytk =_ � 1 _−1_ _[\|][i][ + sinh][ t][j][\|][ d][t]_ � 1 _−1_ [cosh][2][ t][ d][t] = � 1 _−1_ [cosh][ t][ d][t] = [1 + cosh 1 sinh 1] _≈_ 1,197. 2 sinh 1 12. Homogén tömegeloszlású m tömegű vékony drótból a oldalú négyzet alakú keretet hajlítunk. Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatékát. _Megoldás. Legyen a koordinátarendszer olyan, hogy a két átló az x illetve y tengelyre esik._ Ekkor a négy oldal egy-egy paraméterezése (mindegyiknél t ∈ [0, 1]): r(t) = _√[a]_ 2[((1][ −] _[t][)][i][ +][ t][j][)]_ r(t) = _√[a]_ 2[(][−][t][i][ + (1][ −] _[t][)][j][)]_ r(t) = _√[a]_ 2[((][−][1 +][ t][)][i][ −] _[t][j][)]_ r(t) = _√[a]_ 2[(][t][i][ + (][−][1 +][ t][)][j][)] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:49.3pt;left:354.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:59.1pt;left:364.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe ívhossza a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln 2]</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">intervallumon?</span></p> <p style="top:90.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:115.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:103.2pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:127.4pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:144.2pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:131.8pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:156.0pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:133.9pt;left:147.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:133.9pt;left:349.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:144.2pt;left:359.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:133.9pt;left:372.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:172.8pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:160.4pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span></p> <p style="top:184.6pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span></sup></p> <p style="top:181.0pt;left:200.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:196.5pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Mennyi az</span></p> <p style="top:223.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2 sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2 cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:207.8pt;left:250.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:231.9pt;left:260.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup></p> <p style="top:207.8pt;left:300.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:223.7pt;left:309.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:252.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">térgörbe</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterértékeknek megfelelő részének ívhossza?</span></p> <p style="top:268.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:295.5pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:283.0pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></p> <p style="top:307.3pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:295.5pt;left:153.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:325.5pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:313.0pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></p> <p style="top:337.2pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:312.1pt;left:153.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:312.5pt;left:252.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:325.5pt;left:259.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:333.7pt;left:280.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:312.5pt;left:286.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:325.5pt;left:295.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:312.1pt;left:303.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:359.7pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:347.2pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></p> <p style="top:371.4pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:342.0pt;left:153.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">s</span></p> <p style="top:359.7pt;left:163.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:367.9pt;left:185.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:347.2pt;left:234.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></p> <p style="top:371.4pt;left:240.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:346.7pt;left:257.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:359.7pt;left:265.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:367.9pt;left:285.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:346.7pt;left:292.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:359.7pt;left:302.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:343.7pt;left:315.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:367.9pt;left:323.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup></p> <p style="top:343.7pt;left:362.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></p> <p style="top:375.3pt;left:368.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:359.7pt;left:383.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup></p> <p style="top:367.9pt;left:397.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:387.3pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Egy homogén tömegeloszlású vékony kötelet két végénél fogva lógatunk. Az általa meg-</span></p> <p style="top:401.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">határozott görbe egy paraméterezése</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:401.7pt;left:455.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hol van a kötél</span></p> <p style="top:416.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tömegközéppontja?</span></p> <p style="top:433.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A görbe az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelyre szimmetrikus, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A másik komponens az</span></p> <p style="top:447.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elsőrendű nyomaték és a tömeg hányadosa:</span></p> <p style="top:486.5pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:460.2pt;left:139.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:484.5pt;left:144.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:488.5pt;left:153.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:512.7pt;left:159.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:544.2pt;left:125.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:517.9pt;left:139.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:542.1pt;left:144.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:546.1pt;left:141.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:570.4pt;left:147.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:588.4pt;left:125.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + cosh 1 sinh 1</span></sup></p> <p style="top:596.6pt;left:161.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 sinh 1</span></p> <p style="top:588.4pt;left:225.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">197</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:611.8pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Homogén tömegeloszlású</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű vékony drótból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> oldalú négyzet alakú keretet hajlítunk.</span></p> <p style="top:626.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatékát.</span></p> <p style="top:643.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen a koordinátarendszer olyan, hogy a két átló az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> illetve</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelyre esik.</span></p> <p style="top:657.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ekkor a négy oldal egy-egy paraméterezése (mindegyiknél</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">):</span></p> <p style="top:680.6pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup></p> <p style="top:680.0pt;left:142.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:689.7pt;left:152.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">((1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:708.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup></p> <p style="top:708.0pt;left:142.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:717.8pt;left:152.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:736.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup></p> <p style="top:736.1pt;left:142.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:745.8pt;left:152.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">((</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:764.8pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup></p> <p style="top:764.2pt;left:142.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:773.9pt;left:152.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_253.png	9. Mi az r() intervallumon? FALOTT m((.,.m $-sinh 2) 1- (sinh t — cosht) j 4 V3k[de nh 4 coshíji 4. (cosht — sinhí)j 4. Vtk görbe fehossza a ! € f.n2] . Megoldás. 2sin(ti 4 2005(03 b]k . Megoldás. - /_G.UM ű határozott görbe egy pazaméterezőse e) — t 4 coshtj, £ € [-1.11. Hol van a kötől "Megoldás. A görbe az y tengelyre szimmetrikus, tehát 2,. — 0. A másik komponens az elsőrendű nyomaték és a tömeg hányadosa: " cosheJi 4 sinhtjldt AYT 178 coshlsinh 1 2nh 1 n tömegeloszlású m tömegűü vékony drótból a oldalú né v. e 1197. yzet alakú keretet hajlítunk. Határozzuk meg az egyik átlóra vonatkozó tehetetlenségi nyomatókát átló az x illetve y tengelyre csik. t €[D.1). " Megoldás. Legyen a koordinátarendszer olyan, hogy a két Ekkor a egy paraméterezése (mindegyikni r(8 ,%(u 49 oldal egy 10-Se44 a9 4 []
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 4. feladatsor: Felszín, felszíni és felületi integrál 1. Forgassuk meg az y = f(x) diﬀerenciálható függvény graﬁkonját az x tengely körül. Írjuk fel a kapott forgástest egy paraméteres egyenletét. Mekkora az a ≤x ≤b sávba eső rész felszíne? 2. Határozzuk meg az x2 + y2 + z2 = 4 egyenletű gömbfelület (x −1)2 + y2 ≤1 egyenletű hengeren belüli részének a felszínét. 3. Hol van a tömegközéppontja az origó középpontú, vékony, R sugarú, egyenletes µ felületi tömegsűrűségű gömbhéj, z ≥0 féltérbe eső részének? 4. Az r(u, v) = eu cos vi + eu sin vj + uk, u ≤0, v ∈[0, 2π] tölcsér felületi tömegsűrűségét a µ(x, y, z) = 1 + x függvény írja le. Mekkora a tömege? 5. Mi az u(x, y, z) = xi −yj + zk vektormező integrálja az r(u, v) = (u + 2v)i + vj + (u −v)k felület (u, v) ∈[0, 3] × [0, 1] darabján ∂r ∂u ∂v ∂r ∂u × ∂r felület (u, v) ∈[0, 3] × [0, 1] darabján ∂r ∂u × ∂r ∂v irányítás mellett? 6. Határozza meg az u : R3 →R3 u(x, y, z) = (x2 + 2xy −xz)i −y2j −2xzk vektormező integrálját az origó középpontú R = 2 sugarú gömb felületére kifelé mutató irányítás mellett. További gyakorló feladatok 7. Mekkora az r(u, v) = eu(cos vi + sin vj + k) felület (u, v) ∈[0, 1] × [0, π/4] paramétertarto- mánynak megfelelő darabjának a felszíne? 8. Mekkora a felszíne az r : R2 →R3 r(u, v) = ui + vj + uvk felület u2 + v2 ≤1 paramétertartománynak megfelelő darabjának? 9. Számítsuk ki az M tömegű, homogén tömegeloszlású, x2 + y2 = R2, \|z\| ≤ h 2 egyenletű hengerpalást tehetetlenségi nyomatékát a koordinátatengelyekre nézve. 10. Integráljuk a v(x, y, z) = xyi+(2x+z)k vektormezőt az r(u, v) = (u+2v)i−vj+(u2+3v)k felület 0 ≤u ≤3, −2 ≤v ≤0 paramétertartománynak megfelelő darabján ∂r ∂u ∂v ∂r ∂u × ∂r felület 0 ≤u ≤3, −2 ≤v ≤0 paramétertartománynak megfelelő darabján ∂r ∂u × ∂r ∂v irányítás mellett. 11. Integráljuk a v(x, y, z) = (x+y)i+z2k vektormezőt az x2+4y2+9z2 = 1 egyenletű felületen kifelé (a z tengelytől távolodó irányba) mutató irányítás mellett.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 4. feladatsor: Felszín, felszíni és felületi integrál 1. Forgassuk meg az y = f (x) differenciálható függvény grafikonját az x tengely körül. Írjuk fel a kapott forgástest egy paraméteres egyenletét. Mekkora az a ≤ _x ≤_ _b sávba eső rész_ felszíne? 2. Határozzuk meg az x[2] + y[2] + z[2] = 4 egyenletű gömbfelület (x − 1)[2] + y[2] _≤_ 1 egyenletű hengeren belüli részének a felszínét. 3. Hol van a tömegközéppontja az origó középpontú, vékony, R sugarú, egyenletes µ felületi tömegsűrűségű gömbhéj, z ≥ 0 féltérbe eső részének? 4. Az r(u, v) = e[u] cos vi + e[u] sin vj + uk, u ≤ 0, v ∈ [0, 2π] tölcsér felületi tömegsűrűségét a _µ(x, y, z) = 1 + x függvény írja le. Mekkora a tömege?_ 5. Mi az u(x, y, z) = xi − _yj + zk vektormező integrálja az r(u, v) = (u + 2v)i + vj + (u −_ _v)k_ felület (u, v) ∈ [0, 3] × [0, 1] darabján _[∂][r]_ _∂u_ _[×][ ∂]∂v[r]_ [irányítás mellett?] 6. Határozza meg az u : _→_ R[3] R[3] u(x, y, z) = (x[2] + 2xy − _xz)i −_ _y[2]j −_ 2xzk vektormező integrálját az origó középpontú R = 2 sugarú gömb felületére kifelé mutató irányítás mellett. ## További gyakorló feladatok 7. Mekkora az r(u, v) = e[u](cos vi + sin vj + k) felület (u, v) ∈ [0, 1] × [0, π/4] paramétertartománynak megfelelő darabjának a felszíne? 8. Mekkora a felszíne az r : _→_ R[2] R[3] r(u, v) = ui + vj + uvk felület u[2] + v[2] _≤_ 1 paramétertartománynak megfelelő darabjának? 9. Számítsuk ki az M tömegű, homogén tömegeloszlású, x[2] + y[2] = R[2], \|z\| ≤ _h_ 2 [egyenletű] hengerpalást tehetetlenségi nyomatékát a koordinátatengelyekre nézve. 10. Integráljuk a v(x, y, z) = xyi+(2x+z)k vektormezőt az r(u, v) = (u+2v)i−vj+(u[2] +3v)k felület 0 ≤ _u ≤_ 3, −2 ≤ _v ≤_ 0 paramétertartománynak megfelelő darabján _[∂][r]_ _∂u_ _[×][ ∂]∂v[r]_ [irányítás] mellett. 11. Integráljuk a v(x, y, z) = (x+y)i+z[2]k vektormezőt az x[2] +4y[2] +9z[2] = 1 egyenletű felületen kifelé (a z tengelytől távolodó irányba) mutató irányítás mellett. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">4. feladatsor: Felszín, felszíni és felületi integrál</span></b></p> <p style="top:132.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Forgassuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható függvény graﬁkonját az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely körül. Írjuk</span></p> <p style="top:146.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">fel a kapott forgástest egy paraméteres egyenletét. Mekkora az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sávba eső rész</span></p> <p style="top:161.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felszíne?</span></p> <p style="top:177.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű gömbfelület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű</span></p> <p style="top:192.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hengeren belüli részének a felszínét.</span></p> <p style="top:208.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Hol van a tömegközéppontja az origó középpontú, vékony,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú, egyenletes</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> felületi</span></p> <p style="top:223.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tömegsűrűségű gömbhéj,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> féltérbe eső részének?</span></p> <p style="top:239.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tölcsér felületi tömegsűrűségét a</span></p> <p style="top:253.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény írja le. Mekkora a tömege?</span></p> <p style="top:270.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Mi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:284.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> darabján</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:292.1pt;left:270.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:292.1pt;left:297.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">irányítás mellett?</span></sup></p> <p style="top:301.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozza meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">→</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:327.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:354.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektormező integrálját az origó középpontú</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú gömb felületére kifelé mutató</span></p> <p style="top:368.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">irányítás mellett.</span></p> <p style="top:401.3pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:425.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Mekkora az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> felület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paramétertarto-</span></p> <p style="top:439.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mánynak megfelelő darabjának a felszíne?</span></p> <p style="top:456.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Mekkora a felszíne az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">→</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:482.6pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> uv</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:509.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felület</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paramétertartománynak megfelelő darabjának?</span></p> <p style="top:525.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Számítsuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű, homogén tömegeloszlású,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\| ≤</span></i></p> <p style="top:523.9pt;left:479.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i></p> <p style="top:532.8pt;left:479.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletű</span></sup></p> <p style="top:539.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hengerpalást tehetetlenségi nyomatékát a koordinátatengelyekre nézve.</span></p> <p style="top:556.3pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Integráljuk a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:570.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paramétertartománynak megfelelő darabján</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:578.1pt;left:457.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:578.1pt;left:480.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">irányítás</span></sup></p> <p style="top:585.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:601.7pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Integráljuk a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű felületen</span></p> <p style="top:616.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kifelé (a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelytől távolodó irányba) mutató irányítás mellett.</span></p> </div>	page_254.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 4. feladatsor: Felszín, felszíni és felületi integrál vény srafikonját az 7 tengely körül. Írjuk 1. Forgassuk meg az y — f(2) dífferenciálható fi fel a kapott forgástest egy paran felszíne? res egyenletét. Mekkora az a £ r £ b sávba 2. Határozzuk meg az 2? 4 4 4 her 4 egyenletű gömbfelület (z — 172 4-4? £ 1 egyenletű 8. Hol van a tömegközéppontja az origó középpontú, vékony, A sugarú, egyenletes j. felületi tömegsürüsés 20 féltérbe eső részének? 4. Az ríu,v) — eeosvi 4-e"sinvj 4. uk, u £ 0, v € [/,2x] tölcsér fel píz,y.2) — 14r függyény írja le. Mekkora a tömege? ileti tömegsűrűségét a. 5. Mi az u(z.y.2) — zi— yj 4 2k vektormező integrálja az r(u, e) felület (. 2) € [0,3] x [0.1] darabji ellett? 6. Határozza meg az u: R! — R" 4 214-tj4 ú— 1k ulr.y. e 2y vektormező integrálját az origó középpontú R. ányítás mellett sugarú gőmb felületére kifelé n "További gyakorló feladatok 7. Mekkora az rlu,e) — etfeosti 4-sinej - k) felület (u, e) € [0,1] 2e [0, a/4] paran 8. Mekkora a felszíne az r : R? — R? tertartó- KOEE I gerpalást tehetetlenségi nyomatékát a koordinátaton; elyekre nézve. k vektormezőt az r(t. 1) — (4-H2UI—1j4(--30)k ertartománynak megfelelő darabján $r ax 95 10. Integráljuk a V(. y. felület 0 £ u £ 3. mellett. 11. Integráljuk a Vír. y.2) — (z-4-4)i--2?k vektormezőt az 224-4y24-9: kifelé (a : tengelytől távolodó irányba) mutató irányítás mellett. egyenletű felületen
A palást felszíne 2πRh, tehát a felületi tömegsűrűség µ = M 2πRh. A tehetetlenségi nyomatékok számolásához a koordinátafüggvények négyzeteit kell integrálni: Z h/2 −h/2 Z h/2 −h/2 Z h/2 −h/2 Z 2π 0 Z 2π 0 Z 2π 2 R2 cos2 φ · R dφ dz = M 2πRhR3hπ = MR2 2 R2 sin2 φ · R dφ dz = M 2πRhR3hπ = MR2 µx2 dS = M 2πRh µy2 dS = M 2πRh µz2 dS = M 2πRh #h/2 −h/2 "z3 z2 · R dφ dz = M 2πRh2πR = Mh2 12 . = Mh2 12 Két ilyen integrál összege a harmadik tengelyre vonatkozó tehetetlenségi nyomaték, tehát rendre az x, y és z tengelyekre: 1 12M(h2 + 6R2), 1 12M(h2 + 6R2) és MR2. 10. Integráljuk a v(x, y, z) = xyi+(2x+z)k vektormezőt az r(u, v) = (u+2v)i−vj+(u2+3v)k Két ilyen integrál összege a harmadik tengelyre vonatkozó tehetetlenségi nyomaték, tehát rendre az x, y és z tengelyekre: 1 12M(h2 + 6R2), 1 12M(h2 + 6R2) és MR2. felület 0 ≤u ≤3, −2 ≤v ≤0 paramétertartománynak megfelelő darabján ∂r ∂u ∂v ∂r ∂u × ∂r felület 0 ≤u ≤3, −2 ≤v ≤0 paramétertartománynak megfelelő darabján ∂r ∂u × ∂r ∂v irányítás mellett. Megoldás. ∂r ∂u × ∂r ∂v = (i + 2uk) × (2i −j + 3k) = 2ui + (−3 + 4u)j −k és v(r(u, v)) = (−uv −2v2)i + (2u + u2 + 7v)k felhasználásával Z 0 Z 3 v · dA = Z 3 ∂u 0 v(r(u, v)) · ∂r ∂r ∂u × ∂r ∂v ∂r ∂v du dv −2 Z 0 0 Z 3 Z 3 0 ((−uv −2v2)i + (2u + u2 + 7v)k) · (2ui + (−3 + 4u)j −k) du dv −2 Z 0 −2 Z 3 Z 3 0 (−2u −u2 −7v −2u2v −4uv2) du dv = −11. 11. Integráljuk a v(x, y, z) = (x+y)i+z2k vektormezőt az x2+4y2+9z2 = 1 egyenletű felületen kifelé (a z tengelytől távolodó irányba) mutató irányítás mellett. Megoldás. A felületet paraméterezhetjük r(ϑ, ϕ) = sin ϑ cos ϕi + 1 2 3 1 2 sin ϑ sin ϕj + 1 Megoldás. A felületet paraméterezhetjük r(ϑ, ϕ) = sin ϑ cos ϕi + 1 2 sin ϑ sin ϕj + 1 3 cos ϑk módon, ekkor ∂r ∂ϑ × ∂r ∂ϕ = ∂r ∂ϑ × ∂r ∂ϕ cos ϑ cos ϕi + 1 2 1 2 cos ϑ sin ϕj −1 3 1 3 sin ϑk −sin ϑ sin ϕi + 1 2 sin ϑ cos ϕj −sin ϑ sin ϕi + 1 2 1 = sin ϑ 1 6 sin ϑ cos ϕi + 1 3 1 3 sin ϑ sin ϕj + 1 2 1 2 cos ϑ v(r(ϑ, ϕ)) = sin ϑ cos ϕ + 1 2 sin ϑ sin ϕ sin ϑ cos ϕ + 1 2 i + 1 9 cos2 ϑk felhasználásával az integrál: 1 9 cos2 ϑk felhasználásával az integrál: 1 12 sin2 ϑ sin ϕ cos ϕ + cos3 ϑ 18 Z π Z 2π v · dA = 1 6 sin2 ϑ cos2 ϕ + 1 12 18 dϕ dϑ sin2 ϑ n2 ϑ 12 + cos3 ϑ 18 Z π = 2π 18 dϑ = π2 12. dϑ = π2 12	A palást felszíne 2πRh, tehát a felületi tömegsűrűség µ = _M_ 2πRh [. A tehetetlenségi nyomaté-] kok számolásához a koordinátafüggvények négyzeteit kell integrálni: � _M_ � _h/2_ � 2π _M_ _µx[2]_ dS = _R[2]_ cos[2] _φ · R dφ dz =_ 2πRh _−h/2_ 0 2πRh _[R][3][hπ][ =][ MR]2_ [2] � _M_ � _h/2_ � 2π _M_ _µy[2]_ dS = _R[2]_ sin[2] _φ · R dφ dz =_ 2πRh _−h/2_ 0 2πRh _[R][3][hπ][ =][ MR]2_ [2] � _M_ � _h/2_ � 2π _M_ � _z3_ �h/2 _µz[2]_ dS = _z[2]_ _· R dφ dz =_ = _[Mh][2]_ 2πRh _−h/2_ 0 2πRh [2][πR] 3 12 _[.]_ _−h/2_ Két ilyen integrál összege a harmadik tengelyre vonatkozó tehetetlenségi nyomaték, tehát rendre az x, y és z tengelyekre: 1 12 _[M]_ [(][h][2][ + 6][R][2][)][,][ 1]12 _[M]_ [(][h][2][ + 6][R][2][) és][ MR][2][.] 10. Integráljuk a v(x, y, z) = xyi+(2x+z)k vektormezőt az r(u, v) = (u+2v)i−vj+(u[2] +3v)k felület 0 ≤ _u ≤_ 3, −2 ≤ _v ≤_ 0 paramétertartománynak megfelelő darabján _[∂][r]_ _∂u_ _[×][ ∂]∂v[r]_ [irányítás] mellett. _Megoldás._ _∂r_ _∂u_ _[×][ ∂]∂v[r]_ [= (][i][ + 2][u][k][)][ ×][ (2][i][ −] [j][ + 3][k][) = 2][u][i][ + (][−][3 + 4][u][)][j][ −] [k] és v(r(u, v)) = (−uv − 2v[2])i + (2u + u[2] + 7v)k felhasználásával � � 0 v · dA = _−2_ � 0 = _−2_ � 0 = _−2_ � 3 0 [v][(][r][(][u, v][))][ ·][ ∂]∂u[r] _[×][ ∂]∂v[r]_ [d][u][ d][v] � 3 0 [((][−][uv][ −] [2][v][2][)][i][ + (2][u][ +][ u][2][ + 7][v][)][k][)][ ·][ (2][u][i][ + (][−][3 + 4][u][)][j][ −] [k][) d][u][ d][v] � 3 0 [(][−][2][u][ −] _[u][2][ −]_ [7][v][ −] [2][u][2][v][ −] [4][uv][2][) d][u][ d][v][ =][ −][11][.] 11. Integráljuk a v(x, y, z) = (x+y)i+z[2]k vektormezőt az x[2] +4y[2] +9z[2] = 1 egyenletű felületen kifelé (a z tengelytől távolodó irányba) mutató irányítás mellett. _Megoldás. A felületet paraméterezhetjük r(ϑ, ϕ) = sin ϑ cos ϕi +_ [1] 2 [sin][ ϑ][ sin][ ϕ][j][ +][ 1]3 [cos][ ϑ][k] módon, ekkor _∂r_ � � � � _∂ϑ_ _[×][ ∂]∂ϕ[r]_ [=] cos ϑ cos ϕi + [1]2 [cos][ ϑ][ sin][ ϕ][j][ −] [1]3 [sin][ ϑ][k] _×_ _−_ sin ϑ sin ϕi + [1]2 [sin][ ϑ][ cos][ ϕ][j] � 1 � = sin ϑ _._ 6 [sin][ ϑ][ cos][ ϕ][i][ + 1]3 [sin][ ϑ][ sin][ ϕ][j][ + 1]2 [cos][ ϑ] v(r(ϑ, ϕ)) = �sin ϑ cos ϕ + [1] � i + [1] 2 [sin][ ϑ][ sin][ ϕ] 9 [cos][2][ ϑ][k][ felhasználásával az integrál:] � � _π_ � 2π �1 v · dA = 0 0 6 [sin][2][ ϑ][ cos][2][ ϕ][ + 1]12 [sin][2][ ϑ][ sin][ ϕ][ cos][ ϕ][ + cos]18[3][ ϑ] � _π_ �sin2 ϑ � = 2π + [cos][3][ ϑ] dϑ = _[π][2]_ 0 12 18 12 _[.]_ � dϕ dϑ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A palást felszíne</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát a felületi tömegsűrűség</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:57.6pt;left:381.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">M</span></i></p> <p style="top:66.4pt;left:375.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πRh</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tehetetlenségi nyomaté-</span></sup></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kok számolásához a koordinátafüggvények négyzeteit kell integrálni:</span></p> <p style="top:93.9pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:106.4pt;left:118.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µx</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:98.3pt;left:178.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></p> <p style="top:114.6pt;left:170.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i></p> <p style="top:93.9pt;left:202.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:118.1pt;left:207.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:93.9pt;left:230.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:118.1pt;left:235.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:106.4pt;left:251.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:98.3pt;left:369.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></p> <p style="top:114.6pt;left:361.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">hπ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> MR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:114.6pt;left:445.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:126.0pt;left:107.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:138.5pt;left:118.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:130.4pt;left:178.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></p> <p style="top:146.7pt;left:170.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i></p> <p style="top:126.0pt;left:202.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:150.3pt;left:207.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:126.0pt;left:230.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:150.3pt;left:235.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:138.5pt;left:251.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:130.4pt;left:368.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></p> <p style="top:146.7pt;left:360.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">hπ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> MR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:146.7pt;left:444.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:161.3pt;left:107.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:173.8pt;left:119.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:165.7pt;left:178.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></p> <p style="top:182.0pt;left:170.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i></p> <p style="top:161.3pt;left:202.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:185.6pt;left:207.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:161.3pt;left:230.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:185.6pt;left:235.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:173.8pt;left:251.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:165.7pt;left:335.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></p> <p style="top:182.0pt;left:327.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πRh</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πR</span></i></sup></p> <p style="top:157.9pt;left:380.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:182.0pt;left:390.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:157.9pt;left:399.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:189.4pt;left:405.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:173.8pt;left:429.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Mh</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:182.0pt;left:449.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:209.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Két ilyen integrál összege a harmadik tengelyre vonatkozó tehetetlenségi nyomaték, tehát</span></p> <p style="top:223.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rendre az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelyekre:</span></p> <p style="top:222.4pt;left:243.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:231.2pt;left:241.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 6</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:231.2pt;left:324.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 6</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> MR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:243.3pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Integráljuk a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:257.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paramétertartománynak megfelelő darabján</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:265.1pt;left:457.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">r</span></b></sup></p> <p style="top:265.1pt;left:480.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">irányítás</span></sup></p> <p style="top:272.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:291.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:315.2pt;left:108.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:331.5pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:331.5pt;left:138.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup></p> <p style="top:353.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) = (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">uv</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 7</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> felhasználásával</span></p> <p style="top:373.8pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:386.2pt;left:120.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:373.8pt;left:168.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 0</span></p> <p style="top:398.0pt;left:174.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:373.8pt;left:187.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 3</span></p> <p style="top:398.0pt;left:192.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:394.4pt;left:262.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:394.4pt;left:293.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup></p> <p style="top:415.7pt;left:156.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:403.2pt;left:168.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 0</span></p> <p style="top:427.4pt;left:174.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:403.2pt;left:187.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 3</span></p> <p style="top:427.4pt;left:192.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">((</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">uv</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 7</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup></p> <p style="top:445.1pt;left:156.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:432.7pt;left:168.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 0</span></p> <p style="top:456.9pt;left:174.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:432.7pt;left:187.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 3</span></p> <p style="top:456.9pt;left:192.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">uv</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:477.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Integráljuk a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű felületen</span></p> <p style="top:491.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kifelé (a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengelytől távolodó irányba) mutató irányítás mellett.</span></p> <p style="top:510.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A felületet paraméterezhetjük</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:518.1pt;left:420.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:518.1pt;left:499.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup></p> <p style="top:525.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">módon, ekkor</span></p> <p style="top:548.8pt;left:98.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:565.1pt;left:98.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϑ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:565.1pt;left:128.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:544.0pt;left:160.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:556.9pt;left:167.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:565.1pt;left:239.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:565.1pt;left:318.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup></p> <p style="top:544.0pt;left:358.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:556.9pt;left:368.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></p> <p style="top:544.0pt;left:380.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:556.9pt;left:387.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:565.1pt;left:467.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup></p> <p style="top:544.0pt;left:531.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:587.1pt;left:148.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></p> <p style="top:574.1pt;left:185.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:595.3pt;left:194.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:595.3pt;left:273.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:595.3pt;left:351.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup></p> <p style="top:574.1pt;left:385.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:587.1pt;left:394.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:620.9pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span></p> <p style="top:611.0pt;left:143.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:620.9pt;left:149.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:628.2pt;left:215.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup></p> <p style="top:611.0pt;left:272.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:620.9pt;left:280.6pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:628.2pt;left:299.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> felhasználásával az integrál:</span></sup></p> <p style="top:645.3pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:657.8pt;left:120.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:645.3pt;left:168.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:669.6pt;left:174.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:645.3pt;left:186.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:669.6pt;left:191.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:641.9pt;left:208.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:666.0pt;left:217.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:666.0pt;left:301.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup></p> <p style="top:666.0pt;left:422.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">18</span></p> <p style="top:641.9pt;left:444.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:657.8pt;left:454.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i></p> <p style="top:691.7pt;left:156.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:679.2pt;left:183.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:703.4pt;left:189.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:675.7pt;left:201.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></p> <p style="top:699.9pt;left:218.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span></p> <p style="top:691.7pt;left:242.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup></p> <p style="top:699.9pt;left:263.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">18</span></p> <p style="top:675.7pt;left:285.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:691.7pt;left:295.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:699.9pt;left:325.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_255.png	10. 1. A palást felszíne 27.h tel kok számolásához a koordinátafűggy át a felületi tömegsűi k űség p — 327. A tehetetlenségi nyomaté- égyzeteit kell integy Téhz feédsz é [4a h Feo?6-Rdsde - sr E 27Ah rendre az z,y és 2 tengelyekre: L M(N? 4-6A2), L M(H? 4 6R?) és MR. Integráljuk a víz, y.2) — zvi--(22--2)k vektormezőt az r(t,v) — (u-4-201—14j4 11. felület 0 £ u £ 3, —2 £ v £ 0 paramétertartománynak megfelelő darabján $r x 2 mellett . Megoldás. Rdo. k [3 3 (1.4-2uk) x (21—) 4-3k) — 2 4 (-34-409j—k és vír(t. 1) — (—u0 — 24. (24 412 4. TeJk felhasználásával Integráljuk a víz. y.2) — (z4-y)i4-2?k vektormezőt az 2449249. kifelé (a 2 tengelytől távolodó irányba) mutató ír .Megoldás. A fe vezhetjük r(9.s) módon, ekkor egyenletű felületen tet paran 0r ör 1 3 vír(9, )) — (sindeosz 4 $sinvsing)i 4. § cos? Úk felhasználásával az integrál:
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 tavasz 4. feladatsor: Felszín, felszíni és felületi integrál (megoldás) 1. Forgassuk meg az y = f(x) diﬀerenciálható függvény graﬁkonját az x tengely körül. Írjuk fel a kapott forgástest egy paraméteres egyenletét. Mekkora az a ≤x ≤b sávba eső rész felszíne? Megoldás. r(u, v) = ui + f(u) cos vj + f(u) sin vk paraméterezéssel ∂r ∂u × ∂r ∂v ∂r ∂v = (i + f ′(u) cos vj + f ′(u) sin vj) × (−f(u) sin vj + f(u) cos vk) = f(u)f ′(u)i −f(u) cos vj −f(u) sin vk, ebből Z b Z 2π Z b a f(u) Z b ∂r ∂u × ∂r ∂v dv du = 2π A = q 1 + f ′(u) du. 2. Határozzuk meg az x2 + y2 + z2 = 4 egyenletű gömbfelület (x −1)2 + y2 ≤1 egyenletű hengeren belüli részének a felszínét. √ 4 −r2k, ekkor Megoldás. Elég a z ≥0 felét kiszámolni. Legyen a paraméterezés r(r, φ) = r cos φi + r sin φj + ∂r ∂r × ∂r ∂φ ∂r ∂φ = (cos φi + sin φj − r √ 4 −r2) × (−r sin φi + r cos φj) = r2 cos φ √ 4 2 2 cos φ √ 4 −r2i + r2 sin φ √ 4 −r2 sin φ √ 4 −r2j + rk ∂r ∂r × ∂r ∂φ ∂r ∂r × ∂r ∂φ 2r √ 4 −r2 A paramétertartomány φ ∈[−π/2, π/2], r ∈[0, 2 cos φ], tehát Z π/2 −π/2 Z 2 cos φ Z π/2 −π/2 h√ 4 −r2 2r h√ 4 ir=2 cos φ r=0 dφ ir=2 cos φ A = 2 = 8 2r √ 4 −r2 dr dφ = −4 Z π/2 Z π/2 Z / −π/2(1 −\| sin ϕ\|) dϕ = 8π −16 sin ϕ dϕ = 8π −16. 3. Hol van a tömegközéppontja az origó középpontú, vékony, R sugarú, egyenletes µ felületi tömegsűrűségű gömbhéj, z ≥0 féltérbe eső részének? Megoldás. r(ϑ, ϕ) = R sin ϑ cos ϕi + R sin ϑ sin ϕj + R cos ϑk, (ϑ, ϕ) ∈[0, π/2] × [0, 2π] ∂r ∂ϑ × ∂r ∂ϕ = (R cos ϑ cos ϕi + R cos ϑ sin ϕj −R sin ϑk) × (−R sin ϑ sin ϕi + R sin ϑ cos ϕj) = R2 sin2 ϑ cos ϕi + R2 sin2 ϑ sin ϕj + R2 sin ϑ cos ϑk ⇝ ∂r ∂ϑ × ∂r ∂ϕ ∂r ∂ϑ × ∂r ∂ϕ = R2 sin ϑ tömeg: Z 2π Z π/2 µR2 sin ϑ dϑ dϕ = 2πµR2	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 tavasz # 4. feladatsor: Felszín, felszíni és felületi integrál (megoldás) 1. Forgassuk meg az y = f (x) differenciálható függvény grafikonját az x tengely körül. Írjuk fel a kapott forgástest egy paraméteres egyenletét. Mekkora az a ≤ _x ≤_ _b sávba eső rész_ felszíne? _Megoldás. r(u, v) = ui + f_ (u) cos vj + f (u) sin vk paraméterezéssel _∂r_ _∂u_ _[×][ ∂]∂v[r]_ [= (][i][ +][ f][ ′][(][u][) cos][ v][j][ +][ f][ ′][(][u][) sin][ v][j][)][ ×][ (][−][f] [(][u][) sin][ v][j][ +][ f] [(][u][) cos][ v][k][)] = f (u)f _[′](u)i −_ _f_ (u) cos vj − _f_ (u) sin vk, ebből � _b_ � dv du = 2π 1 + f _[′](u) du._ _a_ _[f]_ [(][u][)] �� _b_ _A =_ _a_ � 2π 0 _∂r_ _∂u_ _[×][ ∂]∂v[r]_ �� 2. Határozzuk meg az x[2] + y[2] + z[2] = 4 egyenletű gömbfelület (x − 1)[2] + y[2] _≤_ 1 egyenletű hengeren belüli részének a felszínét. _Megoldás. Elég a z ≥_ 0 felét kiszámolni. Legyen a paraméterezés r(r, φ) = r cos φi + _√_ _r sin φj +_ 4 − _r[2]k, ekkor_ _∂r_ _r_ _∂r_ _[×][ ∂]∂φ[r]_ [= (cos][ φ][i][ + sin][ φ][j][ −] _√4 −_ _r[2]_ [)][ ×][ (][−][r][ sin][ φ][i][ +][ r][ cos][ φ][j][)] = _√[r][2][ cos][ φ]_ _√_ 4 − _r[2]_ [i][ +][ r][2]4[ sin] − _[ φ]r[2]_ [j][ +][ r][k] _∂r_ 2r _∂r_ _[×][ ∂]∂φ[r]_ = _√4 −_ _r[2]_ �� A paramétertartomány φ ∈ [−π/2, π/2], r ∈ [0, 2 cos φ], tehát 4 − _r[2]�r=2 cos φ_ dφ _r=0_ _√_ � � _π/2_ _A = 2_ _−π/2_ � 2 cos φ 2r � _π/2_ _√_ 0 4 − _r[2][ d][r][ d][φ][ =][ −][4]_ _−π/2_ � _π/2_ � _π/2_ = 8 sin ϕ dϕ = 8π − 16. _−π/2[(1][ −\|][ sin][ ϕ][\|][) d][ϕ][ = 8][π][ −]_ [16] 0 3. Hol van a tömegközéppontja az origó középpontú, vékony, R sugarú, egyenletes µ felületi tömegsűrűségű gömbhéj, z ≥ 0 féltérbe eső részének? _Megoldás. r(ϑ, ϕ) = R sin ϑ cos ϕi + R sin ϑ sin ϕj + R cos ϑk, (ϑ, ϕ) ∈_ [0, π/2] × [0, 2π] _∂r_ _∂ϑ_ _[×][ ∂]∂ϕ[r]_ [= (][R][ cos][ ϑ][ cos][ ϕ][i][ +][ R][ cos][ ϑ][ sin][ ϕ][j][ −] _[R][ sin][ ϑ][k][)][ ×][ (][−][R][ sin][ ϑ][ sin][ ϕ][i][ +][ R][ sin][ ϑ][ cos][ ϕ][j][)]_ = R2 sin ϑ �� _∂r_ _∂ϑ_ _[×][ ∂]∂ϕ[r]_ �� tömeg: � 2π 0 = R[2] sin[2] _ϑ cos ϕi + R[2]_ sin[2] _ϑ sin ϕj + R[2]_ sin ϑ cos ϑk ⇝ � _π/2_ _µR[2]_ sin ϑ dϑ dϕ = 2πµR[2] 0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 tavasz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">4. feladatsor: Felszín, felszíni és felületi integrál</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">(megoldás)</span></b></p> <p style="top:149.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Forgassuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható függvény graﬁkonját az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tengely körül. Írjuk</span></p> <p style="top:164.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">fel a kapott forgástest egy paraméteres egyenletét. Mekkora az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sávba eső rész</span></p> <p style="top:178.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felszíne?</span></p> <p style="top:197.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterezéssel</span></p> <p style="top:221.3pt;left:110.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:237.6pt;left:109.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:237.6pt;left:140.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:251.7pt;left:157.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:276.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ebből</span></p> <p style="top:306.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:294.1pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> b</span></i></p> <p style="top:318.4pt;left:136.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></p> <p style="top:294.1pt;left:147.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:318.4pt;left:152.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:290.3pt;left:168.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:298.5pt;left:174.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:314.8pt;left:173.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:314.8pt;left:203.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i></p> <p style="top:290.3pt;left:218.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:294.1pt;left:281.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> b</span></i></p> <p style="top:318.4pt;left:287.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:295.6pt;left:320.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:306.6pt;left:330.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u.</span></i></p> <p style="top:340.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű gömbfelület</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletű</span></p> <p style="top:354.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hengeren belüli részének a felszínét.</span></p> <p style="top:373.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Elég a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> felét kiszámolni.</span></p> <p style="top:373.8pt;left:305.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Legyen a paraméterezés</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:388.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:378.5pt;left:126.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:388.3pt;left:136.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ekkor</span></p> <p style="top:411.1pt;left:116.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:427.4pt;left:116.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂r</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:427.4pt;left:145.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:411.1pt;left:284.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></p> <p style="top:418.6pt;left:267.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:428.3pt;left:276.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:451.6pt;left:164.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup></p> <p style="top:450.9pt;left:177.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:460.7pt;left:187.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup></p> <p style="top:450.9pt;left:238.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:460.7pt;left:248.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup></p> <p style="top:468.0pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:476.3pt;left:113.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:492.5pt;left:113.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂r</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:492.5pt;left:142.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂φ</span></i></p> <p style="top:468.0pt;left:157.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:476.3pt;left:192.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></p> <p style="top:483.7pt;left:177.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:493.5pt;left:187.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:517.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A paramétertartomány</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2 cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:549.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></p> <p style="top:537.1pt;left:140.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:561.3pt;left:146.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:537.1pt;left:169.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2 cos</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> φ</span></i></p> <p style="top:561.3pt;left:174.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:541.5pt;left:220.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></p> <p style="top:548.9pt;left:206.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:558.7pt;left:216.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup></p> <p style="top:537.1pt;left:310.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:561.3pt;left:316.0pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:539.6pt;left:338.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">h</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:549.6pt;left:353.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:539.6pt;left:384.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">i</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">r</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2 cos</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> φ</span></i></p> <p style="top:559.2pt;left:388.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">r</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:549.6pt;left:425.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i></p> <p style="top:581.3pt;left:120.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 8</span></p> <p style="top:568.8pt;left:140.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:593.0pt;left:146.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 8</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span></sup></p> <p style="top:568.8pt;left:302.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:593.0pt;left:308.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:581.3pt;left:328.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:612.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Hol van a tömegközéppontja az origó középpontú, vékony,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú, egyenletes</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> felületi</span></p> <p style="top:626.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tömegsűrűségű gömbhéj,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> féltérbe eső részének?</span></p> <p style="top:645.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span></p> <p style="top:669.4pt;left:87.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:685.7pt;left:87.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϑ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:685.7pt;left:117.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:709.9pt;left:136.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:MSAM10,serif;font-size:12.0pt;color:#000000"> ⇝</span></p> <p style="top:693.6pt;left:418.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:701.8pt;left:423.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:718.1pt;left:422.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϑ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:718.1pt;left:453.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϕ</span></i></p> <p style="top:693.6pt;left:469.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></p> <p style="top:741.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tömeg:</span></p> <p style="top:760.6pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:784.8pt;left:112.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:760.6pt;left:128.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:784.8pt;left:133.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:773.0pt;left:154.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µR</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πµR</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> </div>	page_256.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 tavasz 4. feladatsor: Felszín, felszíni és felületi integrál (megoldás) . Megoldás. r(u. tú 4. f(u)cosej 4 f(u)sin vk paraméterezéssel ör 91 F Uf_:(u Far cos aj 4 f"(a) sin vj) x (—/(u)sinej 4- f(u) cosak) — fedflaji — flu)eoszj — f(u) sin k ebből a-f s Zldcdsas frYETTTŐds v. sgeren belüli részének a felszí 2. Határozzuk meg az 2! h A egyenletű gömbfelület (z — 1) 4 4 £ 1 egyenletű ! Der ö8 (cos ól 4 sin ój x (-rsinól 4-rcosój) c0só , , rösinő 14 L Én$, öx E [7 Va A paramétertartomány 6 € [-7/2.7/2, r € , 205 ], tehát -8 f740-lsineldez sr x:,Á"..,.;(.;:sW 16. tömegsűrűségű gömbhé . Megoldás. (9. e) — Rsin Úcos l 4- Rsin 9sin j 4 Rcos 0k, (9. ) € 10. 2/2] x [0.22] 0r 0r 07 " öz 7 (cosdcosel 4 Reos Vsin j— RsinÚk) x (-Rsindsinyi 4 Rsindcosej) 2 RE sín? 9 cos l 4. R? sin? .sin pj 4 RE sin 0 005 Úk — ] ÁH Á",.um.m.m, mI
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 5. feladatsor: Térfogati integrál 1. Határozzuk meg annak a tórusznak a térfogatát, aminek a középköre R sugarú, a kereszt- metszete pedig r sugarú, r < R. p g g , < 2z 2+ 2+ 2. Az x R y R h 2n ≤1 tartományt homogén anyagú, m tömegű test tölti ki (R, h > 0). Mekkora a koordinátatengelyekre vonatkozó tehetetlenségi nyomatéka? Mi történik, ha n →∞? 3. Mennyi az m tömegű egyenletes tömegeloszlású vékony R sugarú kör alakú drót egy átmé- 2. Az 2+ 2+ 2z rőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot kis keresztmetszetű tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalármező görbementi integrálásával. 4. Számítsuk ki az m tömegű egyenletes tömegeloszlású vékony R sugarú gömbhéj egy átmérő- jére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a gömbhéjat az R+ϵ és R sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek tekintve skalármező felszíni integrálásával. További gyakorló feladatok 5. Mekkora a térfogata az x2 + y2 + z2 ≤4 gömb és a (x −1)2 + y2 ≤1 henger metszetének? (Viviani-féle test) 6. Mennyi az f(x, y, z) = xyz függvény integrálja az x2 + y2 ≤R2, 0 ≤z ≤h hengeren? 7. Integráljuk az f(x, y, z) = √x2 + y2 függvényt az x2 + y2 + z2 ≤R2 gömbön. 8. Integráljuk az f(x, y, z) = xy függvényt az x2 + y2 ≤z2 kúp 0 ≤z ≤2, 0 ≤x, 0 ≤y egyenlőtlenségek által meghatározott darabján. 9. Tekintsük azt a homogén tömegeloszlású tömör tóruszt, amelynek középköre az x−y síkban fekszik, középpontja az origó, a sugara pedig R, és amelynek a keresztmetszete r sugarú (r < R). Hol van a tömegközéppontja a 0 ≤x ≤y egyenlőtlenség által meghatározott darabjának? 10. Számítsuk ki az M tömegű, a élhosszúságú (tömör) szabályos oktaéder tehetetlenségi nyo- matékát a középpontján áthaladó tengelyekre vonatkozóan.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 5. feladatsor: Térfogati integrál 1. Határozzuk meg annak a tórusznak a térfogatát, aminek a középköre R sugarú, a keresztmetszete pedig r sugarú, r < R. 2 2 2n � _x_ � � _y_ � � 2z � 2. Az + + _≤_ 1 tartományt homogén anyagú, m tömegű test tölti ki (R, h > 0). _R_ _R_ _h_ Mekkora a koordinátatengelyekre vonatkozó tehetetlenségi nyomatéka? Mi történik, ha _n →∞?_ 3. Mennyi az m tömegű egyenletes tömegeloszlású vékony R sugarú kör alakú drót egy átmérőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot kis keresztmetszetű tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalármező görbementi integrálásával. 4. Számítsuk ki az m tömegű egyenletes tömegeloszlású vékony R sugarú gömbhéj egy átmérőjére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a gömbhéjat az R+ϵ és R sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek tekintve skalármező felszíni integrálásával. ## További gyakorló feladatok 5. Mekkora a térfogata az x[2] + y[2] + z[2] _≤_ 4 gömb és a (x − 1)[2] + y[2] _≤_ 1 henger metszetének? (Viviani-féle test) 6. Mennyi az f (x, y, z) = xyz függvény integrálja az x[2] + y[2] _≤_ _R[2], 0 ≤_ _z ≤_ _h hengeren?_ 7. Integráljuk az f (x, y, z) = _[√]x[2]_ + y[2] függvényt az x[2] + y[2] + z[2] _≤_ _R[2]_ gömbön. 8. Integráljuk az f (x, y, z) = xy függvényt az x[2] + y[2] _≤_ _z[2]_ kúp 0 ≤ _z ≤_ 2, 0 ≤ _x, 0 ≤_ _y_ egyenlőtlenségek által meghatározott darabján. 9. Tekintsük azt a homogén tömegeloszlású tömör tóruszt, amelynek középköre az x−y síkban fekszik, középpontja az origó, a sugara pedig R, és amelynek a keresztmetszete r sugarú (r < R). Hol van a tömegközéppontja a 0 ≤ _x ≤_ _y egyenlőtlenség által meghatározott_ darabjának? 10. Számítsuk ki az M tömegű, a élhosszúságú (tömör) szabályos oktaéder tehetetlenségi nyomatékát a középpontján áthaladó tengelyekre vonatkozóan. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">5. feladatsor: Térfogati integrál</span></b></p> <p style="top:129.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Határozzuk meg annak a tórusznak a térfogatát, aminek a középköre</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú, a kereszt-</span></p> <p style="top:144.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">metszete pedig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r < R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:163.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Az</span></p> <p style="top:153.4pt;left:93.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:161.8pt;left:101.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:170.7pt;left:101.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">R</span></i></p> <p style="top:153.4pt;left:108.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:153.4pt;left:128.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:161.4pt;left:137.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:170.7pt;left:136.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">R</span></i></p> <p style="top:153.4pt;left:143.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:153.4pt;left:163.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:161.8pt;left:171.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></p> <p style="top:170.7pt;left:172.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i></p> <p style="top:153.4pt;left:180.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tartományt homogén anyagú,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű test tölti ki (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R, h ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">).</span></p> <p style="top:177.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mekkora a koordinátatengelyekre vonatkozó tehetetlenségi nyomatéka? Mi történik, ha</span></p> <p style="top:192.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:208.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Mennyi az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű egyenletes tömegeloszlású vékony</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör alakú drót egy átmé-</span></p> <p style="top:223.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot</span></p> <p style="top:237.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kis keresztmetszetű tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalár-</span></p> <p style="top:252.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mező görbementi integrálásával.</span></p> <p style="top:268.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Számítsuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű egyenletes tömegeloszlású vékony</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú gömbhéj egy átmérő-</span></p> <p style="top:282.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">jére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a gömbhéjat</span></p> <p style="top:297.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϵ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek</span></p> <p style="top:311.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tekintve skalármező felszíni integrálásával.</span></p> <p style="top:344.5pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:368.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Mekkora a térfogata az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gömb és a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> henger metszetének?</span></p> <p style="top:383.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(Viviani-féle test)</span></p> <p style="top:399.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Mennyi az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény integrálja az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> hengeren?</span></p> <p style="top:416.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Integráljuk az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvényt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">gömbön.</span></p> <p style="top:432.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Integráljuk az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvényt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kúp</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:446.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlőtlenségek által meghatározott darabján.</span></p> <p style="top:463.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Tekintsük azt a homogén tömegeloszlású tömör tóruszt, amelynek középköre az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban</span></p> <p style="top:477.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">fekszik, középpontja az origó, a sugara pedig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, és amelynek a keresztmetszete</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú</span></p> <p style="top:492.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r < R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">). Hol van a tömegközéppontja a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenlőtlenség által meghatározott</span></p> <p style="top:506.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">darabjának?</span></p> <p style="top:523.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Számítsuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> élhosszúságú (tömör) szabályos oktaéder tehetetlenségi nyo-</span></p> <p style="top:537.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">matékát a középpontján áthaladó tengelyekre vonatkozóan.</span></p> </div>	page_259.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 5. feladatsor: Térfogati integrál 1. Határozzuk meg annak a tórusznak a térfogatát, aminek a középköre A sugarú, a kereszt metszete pedig r sugarú, r 2 R. £ 1 tartományt homogén anyagú, m tömegű test tölti ki (. A — 0). Mekkora a koordinátatengelyekre vonatkozó tehetetlenségi nyomatéka? Mi történik, ha 3. Mennyi az m tömegű egyenletes tömegelőszlású vé drót egy áti rőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot kis keresztmetszetűi tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalár- mező sörbementi integrálásával. 4. Számútsuk ki az m tömegű egyenletes töm ony A sugai e vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot két Az R--e és R sugarú gömbők közti tekintve skalármező felszíni integrálásával. ony A sugarú kör alaki gömbhéj egy átmi leképp: a gömbbéjat résznek gondolva térfogati interállal illetve felületnek "További gyakorló feladatok 6. Mennyi az /(z. y. S4gömbésa (r-—19244F£1h le test) zy2 függvény integrálja az 2? 4-4? £ R. 0 £ 2 £ h hengeren? 7. Integráljuk az f(r.y.2) — VT F9 függvényt az 2 44 2? £ R? gömbön. 8. Integráljuk az f(r.y.2) — zy függvényt az ? 47 £ Z kúpO s o s20srnoLy egyenlőtlenségek által meghatározott darabján. 9. Tekintsük azt a homogén tömegeloszlású tömör tóruszt, amelynek középköre az 2—y síkban. fekszik, középpontja az origó, a sugara pedig A, és amelynek a keresztmetszete r sugarú (r 2 A). Hol van a tömegközéppontja a 0 £ £ £ y egyenlőtlenség által meghatározott darabjának? 10. Számítsuk ki az M tömegű, a élhosszúság matókát a köz (tömör) szabályos öktaéder tehetetlenségi nyo- ppontján áthaladó tengelyekre vonatkozóan.
Z π/2 π/4 Z 2π Z r 0 ϱ sin ϑρ(R + ρ cos ϑ) dϱ dϑ dϕ = 0. Mz = Z r A tömegközéppont koordinátái: xtk = (2 − √ 2)(4R2 + r2) 2πR 2π √ ytk = 4R2 + r2 2πR ytk √ 2πR ztk = 0. 10. Számítsuk ki az M tömegű, a élhosszúságú (tömör) szabályos oktaéder tehetetlenségi nyo- matékát a középpontján áthaladó tengelyekre vonatkozóan. √ 2 hosszúak. Megoldás. Válasszunk olyan derékszögű koordinátarendszert, amelynek tengelyein helyezkednek el a csúcsok. Ekkor a három tengely cseréjére szimmetrikus az alakzat, tehát elegendő pl. az Mxx és Mxy másodrendű nyomatékokat kiszámolni. Az utóbbi ráadásul 0, hiszen az integrandus előjelet vált az x = 0 síkra való tükrözéskor, az oktaéder pedig önmagába megy át. Az x, y, z ≥0 térnyolcadba az oktaédernek is egynyolcada esik, ez egy tetraéder, amelynek az origóból kiinduló élei páronként merőlegesek és a/ √ tetraéder, amelynek az origóból kiinduló élei páronként merőlegesek és a/ √ 2 hosszúak. Egy ilyen tetraéder térfogata a3 12 2, tehát a sűrűség ϱ = 3M √ 2a3. A tehetetlenségi nyomaték √ 2, tehát a sűrűség ϱ = 3M √ 2a3 dű ték két √ Egy ilyen tetraéder térfogata a3 12 2, tehát a sűrűség ϱ = 3M √ 2a3. A tehetetlenségi nyomaték bármely tengelyre az Mxx másodrendű nyomaték kétszerese, szimmetria miatt ezt elég egy tetraéderre számolni: 12d Ix = Iy = Iz = 2 · 8 · √ Z a/ Z a/ √ 2−x Z a/ √ 2−x−y ϱx2 dz dy dx √ Z a/ √ 0 Z a/ Z a/ √ 2−x x2 dy dx = 16ϱ = 16ϱ a √ 2 −x −y a2 a2 4 −ax √ 2 x √ 2 2 + x2 x2 dx √ = 16ϱ a5 240 √ = 16ϱ a5 240 √ √ 2 = a5ρ 15 2 ρ √ 10 2 = Ma2 Ma 10 .	� _π/2_ _Mz =_ _π/4_ � 2π 0 � _r_ 0 _[ϱ][ sin][ ϑρ][(][R][ +][ ρ][ cos][ ϑ][) d][ϱ][ d][ϑ][ d][ϕ][ = 0][.]_ A tömegközéppont koordinátái: _√_ _xtk = [(2][ −]_ 2)(4R[2] + r[2]) tk 2πR _ytk = [4][R]√[2][ +][ r][2]_ 2πR _ztk = 0._ 10. Számítsuk ki az M tömegű, a élhosszúságú (tömör) szabályos oktaéder tehetetlenségi nyomatékát a középpontján áthaladó tengelyekre vonatkozóan. _Megoldás. Válasszunk olyan derékszögű koordinátarendszert, amelynek tengelyein helyez-_ kednek el a csúcsok. Ekkor a három tengely cseréjére szimmetrikus az alakzat, tehát elegendő pl. az Mxx és Mxy másodrendű nyomatékokat kiszámolni. Az utóbbi ráadásul 0, hiszen az integrandus előjelet vált az x = 0 síkra való tükrözéskor, az oktaéder pedig önmagába megy át. Az x, y, z ≥ 0 térnyolcadba az oktaédernek is egynyolcada esik, ez egy _√_ tetraéder, amelynek az origóból kiinduló élei páronként merőlegesek és a/ 2 hosszúak. Egy ilyen tetraéder térfogata _a√[3]_ _√3M_ 12 2 [, tehát a sűrűség][ ϱ][ =] 2a[3] [. A tehetetlenségi nyomaték] bármely tengelyre az Mxx másodrendű nyomaték kétszerese, szimmetria miatt ezt elég egy tetraéderre számolni: _Ix = Iy = Iz_ _√_ _√_ _√_ � _a/_ 2 � _a/_ 2−x � _a/_ 2−x−y = 2 · 8 · _ϱx[2]_ dz dy dx 0 0 0 _√_ _√_ � _a/_ 2 � _a/_ 2−x � _a_ � = 16ϱ _√_ _x[2]_ dy dx 0 0 2 _[−]_ _[x][ −]_ _[y]_ _√_ � _a/_ 2 �a2 � = 16ϱ _√_ _x[2]_ dx 0 4 _[−]_ _[ax]2 [+][ x]2[2]_ _a[5]_ = 16ϱ _√_ 240 _a[5]ρ_ _√_ 2 [=] 15 2 [=][ Ma]10 [2] _[.]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:64.0pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:51.5pt;left:140.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:75.8pt;left:145.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></p> <p style="top:51.5pt;left:166.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:75.8pt;left:171.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:51.5pt;left:188.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> r</span></i></p> <p style="top:75.8pt;left:193.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑρ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ρ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:90.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A tömegközéppont koordinátái:</span></p> <p style="top:124.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:106.5pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:116.3pt;left:173.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)(4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:132.6pt;left:176.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πR</span></i></p> <p style="top:154.1pt;left:107.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:153.5pt;left:144.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:163.2pt;left:154.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πR</span></i></p> <p style="top:179.0pt;left:107.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">tk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:205.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Számítsuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> élhosszúságú (tömör) szabályos oktaéder tehetetlenségi nyo-</span></p> <p style="top:219.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">matékát a középpontján áthaladó tengelyekre vonatkozóan.</span></p> <p style="top:239.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Válasszunk olyan derékszögű koordinátarendszert, amelynek tengelyein helyez-</span></p> <p style="top:253.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kednek el a csúcsok. Ekkor a három tengely cseréjére szimmetrikus az alakzat, tehát ele-</span></p> <p style="top:268.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">gendő pl. az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">xx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> másodrendű nyomatékokat kiszámolni. Az utóbbi ráadásul</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:282.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hiszen az integrandus előjelet vált az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkra való tükrözéskor, az oktaéder pedig ön-</span></p> <p style="top:297.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">magába megy át. Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x, y, z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> térnyolcadba az oktaédernek is egynyolcada esik, ez egy</span></p> <p style="top:311.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tetraéder, amelynek az origóból kiinduló élei páronként merőlegesek és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a/</span></i></p> <p style="top:301.8pt;left:468.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:311.5pt;left:478.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> hosszúak.</span></p> <p style="top:326.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Egy ilyen tetraéder térfogata</span></p> <p style="top:324.5pt;left:238.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup></p> <p style="top:334.1pt;left:232.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span></p> <p style="top:327.5pt;left:241.0pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:334.1pt;left:248.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát a sűrűség</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:324.5pt;left:373.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">M</span></i></p> <p style="top:327.5pt;left:370.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:334.1pt;left:377.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tehetetlenségi nyomaték</span></sup></p> <p style="top:340.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">bármely tengelyre az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">xx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> másodrendű nyomaték kétszerese, szimmetria miatt ezt elég egy</span></p> <p style="top:354.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tetraéderre számolni:</span></p> <p style="top:377.9pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> I</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> I</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></p> <p style="top:404.5pt;left:122.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 8</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></p> <p style="top:392.0pt;left:163.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a/</span></i></p> <p style="top:390.2pt;left:182.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:396.8pt;left:189.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:416.3pt;left:169.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:392.0pt;left:196.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a/</span></i></p> <p style="top:390.2pt;left:214.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:396.8pt;left:221.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:416.3pt;left:201.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:392.0pt;left:239.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a/</span></i></p> <p style="top:390.2pt;left:258.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:396.8pt;left:265.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:416.3pt;left:245.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:404.5pt;left:294.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱx</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:435.3pt;left:122.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱ</span></i></p> <p style="top:422.9pt;left:154.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a/</span></i></p> <p style="top:421.0pt;left:173.0pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:427.6pt;left:180.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:447.1pt;left:159.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:422.9pt;left:186.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a/</span></i></p> <p style="top:421.0pt;left:205.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:427.6pt;left:212.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:447.1pt;left:192.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:419.4pt;left:230.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></p> <p style="top:434.7pt;left:239.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:444.4pt;left:249.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:419.4pt;left:298.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:435.3pt;left:308.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:469.5pt;left:122.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱ</span></i></p> <p style="top:457.0pt;left:154.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a/</span></i></p> <p style="top:455.2pt;left:173.0pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:461.8pt;left:180.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:481.3pt;left:159.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:453.6pt;left:186.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:477.7pt;left:198.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ax</span></i></sup></p> <p style="top:468.9pt;left:223.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:478.6pt;left:233.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:477.7pt;left:259.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:453.6pt;left:269.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:469.5pt;left:278.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:503.0pt;left:122.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱ</span></i></p> <p style="top:494.9pt;left:164.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup></p> <p style="top:512.1pt;left:153.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">240</span></p> <p style="top:502.4pt;left:171.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:512.1pt;left:181.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:494.9pt;left:210.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ</span></i></p> <p style="top:512.1pt;left:205.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15</span></p> <p style="top:502.4pt;left:216.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:512.1pt;left:226.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ma</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:511.2pt;left:256.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_260.png	10. a [1Ű [T [/os00(R- peosW)dodúde — 0. A töm. özéppont koordinátái: - vVöaR 3) n 2R WVA a0. Számítsuk ki az M tömegű, a élhosszúságú (tömör) szabályos öktaéder tehetetlenségi nyo- matókát a középpontján áthaladó tengelyekre vonatkozóan. Egy ilven tetraéder térfogata :£. tehái A tehetetlenségi nyomaték 1.
3. Mennyi az m tömegű egyenletes tömegeloszlású vékony R sugarú kör alakú drót egy átmé- rőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot kis keresztmetszetű tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalármező görbementi integrálásával. Megoldás. [I] A drótot olyan tórusznak gondolhatjuk, aminek középköre R sugarú, keresztmetszete pedig ε ≪R sugarú. A tórusz egy paraméterezése r(r, ϑ, ϕ) = (R+r cos ϑ) cos ϕi+ (R+r cos ϑ) sin ϕj+r sin ϑk, ahol (r, ϑ, ϕ) ∈[0, ε]×[0, 2π]×[0, 2π]. A paraméterezés Jacobideterminánsa r(R + r cos ϑ). Ha a tórusz sűrűsége ϱ, akkor tömege m = ϱV = ϱ · 2π2Rε2, tehát ϱ = m 2π2Rε2. A tehetetlenségi nyomaték: Z ε Z 2π Z 2π 0 Z 2π ϱ((R + r cos ϑ)2 cos2 ϕ + r2 sin2 ϑ)r(R + r cos ϑ) dϕ dϑ dr = ϱ 0Z ε Z 2π 0 (rR3 cos2 ϕ + 3r2R2 cos ϑ cos2 ϕ + 3r3R cos2 ϑ cos2 ϕ + r4 cos3 ϑ cos2 ϕ 0 0 0 + r3R sin2 ϕ + r4 cos ϑ sin2 ϕ) dϕ dϑ dr = ϱ1 4 1 4π2ε2(4R3 + 5Rε2) = 1 8 1 8m(4R2 + 5ε2) →mR2 2 Megoldás. [II] A drótót körvonalnak is gondolhatjuk, ekkor a sűrűsége végtelen, de lehet vonalmenti sűrűségről beszélni: m = µ · 2πR, tehát µ = m 2πR. A körvonal legyen az x-y síkban, egy paraméterezés r(ϕ) = R cos ϕi + R sin ϕj. A tehetetlenségi nyomaték: Z 2π µR2 cos2 ϕR dϕ = µR3π = mR2 2 4. Számítsuk ki az m tömegű egyenletes tömegeloszlású vékony R sugarú gömbhéj egy átmérő- jére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a gömbhéjat az R+ϵ és R sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek tekintve skalármező felszíni integrálásával. Megoldás. [I] A gömbhéjra úgy gondolhatunk, hogy az origótól legalább R és legfeljebb R + ϵ távolságra lévő pontok halmaza, ahol ϵ ≪R. Ha a sűrűsége ρ, akkor a tömege m = ρ · 4 3π[(R + ϵ)3 −R3]. A másodrendű nyomaték minden irányban egyforma, tehát a tehetetlenségi nyomaték ennek kétszerese. Gömbi koordinátákkal számolva Z 2π Z π Z R+ϵ Z π 0 cos2 ϑ sin ϑ dϑ I = 2 ρ(r cos ϑ)2r2 sin ϑ dr dϑ dϕ = 4πρ Z π Z R+ϵ r4 dr = 2m 5 0 R (R + ϵ)5 −R5 (R + ϵ)5 −R5 3 (R + ϵ)3 −R3 →2 2 3mR2 Megoldás. [II] A gömbhéjat felületnek idealizálva a sűrűség végtelenné válik, de beszélhetünk felületi sűrűségről: m = µ · 4πR2, tehát µ = m 4πR2. r(ϑ, ϕ) = R(sin ϑ cos ϕi + sin ϑ sin ϕj + cos ϑk) paraméterezéssel Z 2π Z π 0 µ(R cos ϑ)2 Z π Z π ∂r ∂ϑ × ∂r ∂ϕ ∂r ∂ϑ × ∂r ∂ϕ dϑ dϕ = 4πR2µ Z π 3 0 cos2 ϑ sin ϑ dϑ = 2 2 3mR2 I = 2 További gyakorló feladatok 5. Mekkora a térfogata az x2 + y2 + z2 ≤4 gömb és a (x −1)2 + y2 ≤1 henger metszetének? (Viviani-féle test)	3. Mennyi az m tömegű egyenletes tömegeloszlású vékony R sugarú kör alakú drót egy átmérőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot kis keresztmetszetű tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalármező görbementi integrálásával. _Megoldás. [I] A drótot olyan tórusznak gondolhatjuk, aminek középköre R sugarú, kereszt-_ metszete pedig ε ≪ _R sugarú. A tórusz egy paraméterezése r(r, ϑ, ϕ) = (R+r cos ϑ) cos ϕi+_ (R+r cos ϑ) sin ϕj+r sin ϑk, ahol (r, ϑ, ϕ) ∈ [0, ε]×[0, 2π]×[0, 2π]. A paraméterezés Jacobideterminánsa r(R + r cos ϑ). Ha a tórusz sűrűsége ϱ, akkor tömege m = ϱV = ϱ · 2π[2]Rε[2], tehát ϱ = _m_ 2π[2]Rε[2] [. A tehetet-] lenségi nyomaték: � _ε_ � 2π � 2π _ϱ((R + r cos ϑ)[2]_ cos[2] _ϕ + r[2]_ sin[2] _ϑ)r(R + r cos ϑ) dϕ dϑ dr_ 0 0 0 � _ε_ � 2π � 2π = ϱ (rR[3] cos[2] _ϕ + 3r[2]R[2]_ cos ϑ cos[2] _ϕ + 3r[3]R cos[2]_ _ϑ cos[2]_ _ϕ + r[4]_ cos[3] _ϑ cos[2]_ _ϕ_ 0 0 0 + r[3]R sin[2] _ϕ + r[4]_ cos ϑ sin[2] _ϕ) dϕ dϑ dr_ = ϱ [1] 4[π][2][ε][2][(4][R][3][ + 5][Rε][2][) = 1]8[m][(4][R][2][ + 5][ε][2][)][ →] _[mR]2_ [2] _Megoldás. [II] A drótót körvonalnak is gondolhatjuk, ekkor a sűrűsége végtelen, de lehet_ vonalmenti sűrűségről beszélni: m = µ · 2πR, tehát µ = _m_ 2πR [. A körvonal legyen az][ x][-][y] síkban, egy paraméterezés r(ϕ) = R cos ϕi + R sin ϕj. A tehetetlenségi nyomaték: � 2π _µR[2]_ cos[2] _ϕR dϕ = µR[3]π =_ _[mR][2]_ 0 2 4. Számítsuk ki az m tömegű egyenletes tömegeloszlású vékony R sugarú gömbhéj egy átmérőjére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a gömbhéjat az R+ϵ és R sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek tekintve skalármező felszíni integrálásával. _Megoldás. [I] A gömbhéjra úgy gondolhatunk, hogy az origótól legalább R és legfeljebb_ _R + ϵ távolságra lévő pontok halmaza, ahol ϵ ≪_ _R. Ha a sűrűsége ρ, akkor a tömege_ _m = ρ ·_ [4] 3 _[π][[(][R][ +][ ϵ][)][3][ −]_ _[R][3][]. A másodrendű nyomaték minden irányban egyforma, tehát a]_ tehetetlenségi nyomaték ennek kétszerese. Gömbi koordinátákkal számolva � 2π � _π_ � _R+ϵ_ � _π_ � _R+ϵ_ _I = 2_ _ρ(r cos ϑ)[2]r[2]_ sin ϑ dr dϑ dϕ = 4πρ _r[4]_ dr 0 0 _R_ 0 [cos][2][ ϑ][ sin][ ϑ][ d][ϑ] _R_ (R + ϵ)[5] _−_ _R[5]_ = [2][m] 5 (R + ϵ)[3] _−_ _R[3][ →]_ [2]3[mR][2] _Megoldás. [II] A gömbhéjat felületnek idealizálva a sűrűség végtelenné válik, de beszél-_ hetünk felületi sűrűségről: m = µ · 4πR[2], tehát µ = _m_ r(ϑ, ϕ) = R(sin ϑ cos ϕi + 4πR[2] [.] sin ϑ sin ϕj + cos ϑk) paraméterezéssel dϑ dϕ = 4πR2µ � _π_ �� 0 [cos][2][ ϑ][ sin][ ϑ][ d][ϑ][ = 2]3[mR][2] � 2π _I = 2_ 0 � _π_ _∂r_ 0 _[µ][(][R][ cos][ ϑ][)][2]_ _∂ϑ_ _[×][ ∂]∂ϕ[r]_ �� ## További gyakorló feladatok 5. Mekkora a térfogata az x[2] + y[2] + z[2] _≤_ 4 gömb és a (x − 1)[2] + y[2] _≤_ 1 henger metszetének? (Viviani-féle test) -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Mennyi az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű egyenletes tömegeloszlású vékony</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú kör alakú drót egy átmé-</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot</span></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kis keresztmetszetű tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalár-</span></p> <p style="top:102.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mező görbementi integrálásával.</span></p> <p style="top:121.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [I] A drótot olyan tórusznak gondolhatjuk, aminek középköre</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú, kereszt-</span></p> <p style="top:136.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">metszete pedig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ε</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≪</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú. A tórusz egy paraméterezése</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:150.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, ε</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A paraméterezés Jacobi-</span></p> <p style="top:165.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">determinánsa</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:179.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha a tórusz sűrűsége</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor tömege</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱV</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Rε</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:178.1pt;left:457.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></p> <p style="top:186.9pt;left:447.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Rε</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tehetet-</span></sup></p> <p style="top:194.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lenségi nyomaték:</span></p> <p style="top:215.9pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> ε</span></i></p> <p style="top:240.1pt;left:114.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:215.9pt;left:124.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:240.1pt;left:130.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:215.9pt;left:146.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:240.1pt;left:152.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:228.4pt;left:168.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϱ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">((</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></p> <p style="top:257.0pt;left:109.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱ</span></i></p> <p style="top:244.5pt;left:130.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> ε</span></i></p> <p style="top:268.8pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:244.5pt;left:146.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:268.8pt;left:152.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:244.5pt;left:168.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:268.8pt;left:174.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:257.0pt;left:188.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">rR</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i></p> <p style="top:279.7pt;left:109.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></p> <p style="top:305.5pt;left:269.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϱ</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:313.7pt;left:290.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ε</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Rε</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></sup></p> <p style="top:313.7pt;left:405.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ε</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">mR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:313.7pt;left:512.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:335.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [II] A drótót körvonalnak is gondolhatjuk, ekkor a sűrűsége végtelen, de lehet</span></p> <p style="top:350.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vonalmenti sűrűségről beszélni:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πR</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:348.6pt;left:383.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></p> <p style="top:357.4pt;left:379.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πR</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A körvonal legyen az</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:364.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">síkban, egy paraméterezés</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tehetetlenségi nyomaték:</span></p> <p style="top:385.6pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:409.8pt;left:112.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:398.1pt;left:128.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µR</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕR</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µR</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">mR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:406.3pt;left:275.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:427.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Számítsuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tömegű egyenletes tömegeloszlású vékony</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú gömbhéj egy átmérő-</span></p> <p style="top:442.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">jére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a gömbhéjat</span></p> <p style="top:456.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϵ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek</span></p> <p style="top:471.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tekintve skalármező felszíni integrálásával.</span></p> <p style="top:490.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [I] A gömbhéjra úgy gondolhatunk, hogy az origótól legalább</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és legfeljebb</span></p> <p style="top:505.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϵ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> távolságra lévő pontok halmaza, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϵ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≪</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ha a sűrűsége</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ρ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor a tömege</span></p> <p style="top:519.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ρ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:526.8pt;left:121.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϵ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A másodrendű nyomaték minden irányban egyforma, tehát a</span></sup></p> <p style="top:533.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehetetlenségi nyomaték ennek kétszerese. Gömbi koordinátákkal számolva</span></p> <p style="top:565.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></p> <p style="top:553.1pt;left:138.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:577.3pt;left:143.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:553.1pt;left:160.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:577.3pt;left:165.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:553.1pt;left:177.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ϵ</span></i></p> <p style="top:577.3pt;left:183.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">R</span></i></p> <p style="top:565.6pt;left:206.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ρ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πρ</span></i></p> <p style="top:553.1pt;left:376.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:577.3pt;left:382.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i></sup></p> <p style="top:553.1pt;left:466.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ϵ</span></i></p> <p style="top:577.3pt;left:471.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">R</span></i></p> <p style="top:565.6pt;left:495.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></p> <p style="top:595.7pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i></sup></p> <p style="top:603.9pt;left:136.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:587.6pt;left:150.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϵ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup></p> <p style="top:603.9pt;left:150.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϵ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:603.9pt;left:241.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">mR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:628.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [II] A gömbhéjat felületnek idealizálva a sűrűség végtelenné válik, de beszél-</span></p> <p style="top:643.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hetünk felületi sűrűségről:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πR</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> µ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:641.7pt;left:374.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></p> <p style="top:650.5pt;left:367.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πR</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:643.2pt;left:402.3pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ, ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> R</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:657.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterezéssel</span></p> <p style="top:691.5pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></p> <p style="top:679.1pt;left:138.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:703.3pt;left:143.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:679.1pt;left:160.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:703.3pt;left:165.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">R</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:675.2pt;left:236.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:683.5pt;left:241.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:699.7pt;left:240.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϑ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:699.7pt;left:271.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂ϕ</span></i></p> <p style="top:675.2pt;left:286.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πR</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">µ</span></i></p> <p style="top:679.1pt;left:373.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> π</span></i></p> <p style="top:703.3pt;left:378.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϑ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></sup></p> <p style="top:699.7pt;left:477.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">mR</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:729.1pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:753.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Mekkora a térfogata az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gömb és a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> henger metszetének?</span></p> <p style="top:767.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(Viviani-féle test)</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_262.png	3. Mennyi az m tömegű egye drót egy átmó- rőjére vonatkozó tehetetlenségi nyomatéka? Oldjuk meg a feladatot kétféleképp: a drótot kis keresztmetszetűi tórusznak tekintve térfogati integrállal illetve vonalnak tekintve skalár- mező sörbementi integrálásával. letes tömegeloszlású vékony A sugarú kör alaki . Megoldás. ] A drótot olyan tórusznak gondolhatjuk, aminek középköre A sugarú, keres metszete pedig e ££ Fsugarú. A tórusz egy paraméterezése rír, . ) — (R--r-cos V) cos l-. (R--rcosd)sin ej-ersin Úk, ahol (r. 9. ) € [9. e[ x [0.271x[9.221. A paraméterezés Jacobi- determinánsa r(R 4-rcos). irűsége 0. akkor töm nyomaté K [Ő oR reosüfcség r V — 9. 252Re, tehát 9 — szAez. A tehetet- s gyelR 4 recsú) dpdddr Fr Rsint p rteos úsin? g)dedődr 1rel4R 4 5 1442 A aR 4 .Megoldás. M) A drótót körvonalnak is gondolhatjuk, ekkor a sűrűsége végtelen, de lehet ről beszélni: m — j . 27, tehát j — -2x. A körvonal legyen az 2-y Reosgi 4. Rsins. A teketetlenségi nyomaték: vonalmenti síkban, egy paraméterezés r() mR [7 sttcssotás etőn - A. Számútsuk ki az m tömegi jére vonatkozó tehetetlenségi nyomatékát. Oldjuk meg a feladatot kétféleképp: a görmbhéjat Az R4-e és R sugarú gömbök közti térrésznek gondolva térfogati integrállal illetve felületnek tekintve skalármező fels .Megoldás. 1) A sömbbé R -. e távolságra lévő pontok hi íni integrálásával. a úgy sondolhatunk, hogy az origótól legalább 1 és legfeljebb lmaza, ahol e £ R. Ha a sűrűsége p, akkor a tömege 2m(R-P -Ő RTB né válik, de beszél- R(sin V cos e 4 .Megoldás. (II] A sömbhéjat felületnek idealizálva a sűrűség végte hetünk felületi sűrűségről. m. A Azf, tehát a — 3a. rd.s) 27 [ ráítoss9y "További gyakorló feladatok (Vivinni-féle test) ör Ezi BE aydíe — arfén [ cosAsinddd — %mr' E S4gömbésa (r—194-£1h
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 6. feladatsor: Integrálátalakító tételek 1. Mennyi az u(x, y, z) = x(x −2xy + 2yz2)i −y(2x2 + 4xyz + yz2)j + 2xz(x + 2y + 2yz)k vektormező integrálja a 0 ≤x ≤1, 0 ≤y ≤1, 0 ≤z ≤1 egységkocka felületén kifelé mutató irányítás mellett? 2. Határozzuk meg az u(x, y, z) = −xzi + (−xy + 2xz −yz)j + (x2 + xz + z2)k vektormező integrálját azon az irányított felületen, amely az x2 + y2 ≤1, 0 ≤z ≤10 −x egyenlőtlenségek által meghatározott tartomány kifelé irányított felületéből a z = 0 síkba eső alapkör elhagyásával keletkezik. 3. Integráljuk az u(x, y, z) = y2i + z2j + x2k vektormezőt az ABC háromszögvonalon (ebben az irányban), ahol A = (a, 0, 0), B = (0, a, 0) és C = (0, 0, a), a > 0. 4. Integráljuk a v(x, y, z) = −x2yi+xy2j vektormezőt az x2 +y2 = a2 egyenletű körön pozitív forgásiránnyal. További gyakorló feladatok 5. Számítsuk ki az u(x, y, z) = yzi + xzj + xyk vektormező integrálját az x ≥0, y ≥0, z ≥ 0, x + y + z ≤3 tetraéder felületén kifelé mutató irányítás mellett. 6. Mennyi az u(x, y, z) = (xy + 5z2)i + y2j + (xz −y2)k vektormező integrálja annak a tetra- édernek a felületén kifelé mutató irányítás mellett, amelynek csúcsai (1, −2, 3), (0, −1, 1), (0, −3, 0), (4, 3, 3)? 7. Mennyi az u(x, y, z) = (xy + yz)i + (x2 −yz)j + (2xy + z2)k vektormező integrálja az ti + t2j ha t ∈[0, 1] i + j + (t −1)k ha t ∈[1, 2] (3 −t)i + (3 −t)2j + k ha t ∈[2, 3] (4 −t)k ha t ∈[3, 4] r(t) = görbe t ∈[0, 4] darabján? 8. Határozzuk meg az u(x, y, z) = (xy2 −y2z + x2)i + (x2y −xyz)j + (yz2 −x2z)k vektormező g ( , y, ) integrálját az r(t) = (t3 + t)i + √ g ( , y, ) ( y y ) ( y y )j (y ) integrálját az r(t) = (t3 + t)i + 4 + 3t2 −t4j görbe t ∈[−2, 2] intervallumnak megfelelő szakaszán. 9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol f skalármező, u vektormező, S peremes irányított felület: Z S(f rot u) · dA = Z ∂S fu · dr − Z S(grad f × u) · dA 10. Bizonyítsuk be, hogy egy T síkidom területét A = I ∂T xj · dr = − I ∂T yi · dr módon is számíthatjuk és ennek segítségével határozzuk meg az r(t) = cos3 ti + sin3 tj görbével határolt asztroid területét.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 6. feladatsor: Integrálátalakító tételek 1. Mennyi az u(x, y, z) = x(x − 2xy + 2yz[2])i − _y(2x[2]_ + 4xyz + yz[2])j + 2xz(x + 2y + 2yz)k vektormező integrálja a 0 ≤ _x ≤_ 1, 0 ≤ _y ≤_ 1, 0 ≤ _z ≤_ 1 egységkocka felületén kifelé mutató irányítás mellett? 2. Határozzuk meg az u(x, y, z) = −xzi + (−xy + 2xz − _yz)j + (x[2]_ + xz + z[2])k vektormező integrálját azon az irányított felületen, amely az x[2] + y[2] _≤_ 1, 0 ≤ _z ≤_ 10 − _x egyenlőtlen-_ ségek által meghatározott tartomány kifelé irányított felületéből a z = 0 síkba eső alapkör elhagyásával keletkezik. 3. Integráljuk az u(x, y, z) = y[2]i + z[2]j + x[2]k vektormezőt az ABC háromszögvonalon (ebben az irányban), ahol A = (a, 0, 0), B = (0, a, 0) és C = (0, 0, a), a > 0. 4. Integráljuk a v(x, y, z) = −x[2]yi + _xy[2]j vektormezőt az x[2]_ + _y[2]_ = a[2] egyenletű körön pozitív forgásiránnyal. ## További gyakorló feladatok 5. Számítsuk ki az u(x, y, z) = yzi + xzj + xyk vektormező integrálját az x ≥ 0, y ≥ 0, z ≥ 0, x + y + z ≤ 3 tetraéder felületén kifelé mutató irányítás mellett. 6. Mennyi az u(x, y, z) = (xy + 5z[2])i + y[2]j + (xz − _y[2])k vektormező integrálja annak a tetra-_ édernek a felületén kifelé mutató irányítás mellett, amelynek csúcsai (1, −2, 3), (0, −1, 1), (0, −3, 0), (4, 3, 3)? 7. Mennyi az u(x, y, z) = (xy + yz)i + (x[2] _−_ _yz)j + (2xy + z[2])k vektormező integrálja az_ r(t) = ti + t[2]j ha t ∈ [0, 1] i + j + (t − 1)k ha t ∈ [1, 2] (3 − _t)i + (3 −_ _t)[2]j + k_ ha t ∈ [2, 3] (4 − _t)k_ ha t ∈ [3, 4] görbe t ∈ [0, 4] darabján? 8. Határozzuk meg az u(x, y, z) = (xy[2] _−_ _y[2]z + x[2])i + (x[2]y −_ _xyz)j + (yz[2]_ _−_ _x[2]z)k vektormező_ _√_ integrálját az r(t) = (t[3] + t)i + 4 + 3t[2] _−_ _t[4]j görbe t ∈_ [−2, 2] intervallumnak megfelelő szakaszán. 9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol f skalármező, u vektormező, _S peremes irányított felület:_ � � � _S[(][f][ rot][ u][)][ ·][ d][A][ =]_ _∂S_ _[f]_ [u][ ·][ d][r][ −] _S[(grad][ f][ ×][ u][)][ ·][ d][A]_ 10. Bizonyítsuk be, hogy egy T síkidom területét � � _A =_ _∂T_ _[x][j][ ·][ d][r][ =][ −]_ _∂T_ _[y][i][ ·][ d][r]_ módon is számíthatjuk és ennek segítségével határozzuk meg az r(t) = cos[3] _ti + sin[3]_ _tj_ görbével határolt asztroid területét. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">6. feladatsor: Integrálátalakító tételek</span></b></p> <p style="top:130.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:145.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektormező integrálja a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egységkocka felületén kifelé</span></p> <p style="top:159.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mutató irányítás mellett?</span></p> <p style="top:176.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező</span></p> <p style="top:190.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">integrálját azon az irányított felületen, amely az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenlőtlen-</span></p> <p style="top:205.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ségek által meghatározott tartomány kifelé irányított felületéből a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkba eső alapkör</span></p> <p style="top:219.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elhagyásával keletkezik.</span></p> <p style="top:236.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Integráljuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ABC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> háromszögvonalon (ebben</span></p> <p style="top:250.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az irányban), ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, a,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:266.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Integráljuk a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletű körön pozitív</span></p> <p style="top:281.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">forgásiránnyal.</span></p> <p style="top:314.1pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:338.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Számítsuk ki az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálját az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i></p> <p style="top:352.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tetraéder felületén kifelé mutató irányítás mellett.</span></p> <p style="top:369.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja annak a tetra-</span></p> <p style="top:383.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">édernek a felületén kifelé mutató irányítás mellett, amelynek csúcsai</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:397.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:414.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span></p> <p style="top:468.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:433.7pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:442.7pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:445.7pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:448.6pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:451.6pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:454.6pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:457.6pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:475.5pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:478.5pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:481.5pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:484.5pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:487.5pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:490.5pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:442.5pt;left:149.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:442.5pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span></p> <p style="top:459.8pt;left:149.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:459.8pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span></p> <p style="top:477.1pt;left:149.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b></p> <p style="top:477.1pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3]</span></p> <p style="top:494.5pt;left:149.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:494.5pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4]</span></p> <p style="top:521.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbe</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> darabján?</span></p> <p style="top:538.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező</span></p> <p style="top:552.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">integrálját az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:542.7pt;left:242.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:552.5pt;left:252.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> intervallumnak megfelelő</span></p> <p style="top:566.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szakaszán.</span></p> <p style="top:583.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> skalármező,</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező,</span></p> <p style="top:597.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> peremes irányított felület:</span></p> <p style="top:614.7pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:638.9pt;left:112.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> rot</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:614.7pt;left:201.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:638.9pt;left:207.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂S</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:614.7pt;left:270.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:638.9pt;left:275.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(grad</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup></p> <p style="top:656.9pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Bizonyítsuk be, hogy egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> T</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkidom területét</span></p> <p style="top:686.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:673.8pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">I</span></p> <p style="top:698.0pt;left:136.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂T</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:673.8pt;left:208.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">I</span></p> <p style="top:698.0pt;left:213.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂T</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:717.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">módon is számíthatjuk és ennek segítségével határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:731.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbével határolt asztroid területét.</span></p> </div>	page_264.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 6. feladatsor: Integrálátalakító tételek 1. Mennyi az ülr,y.2) — 2(r — 2xy 4 2y22)i — y(2r? 4 deye 4 y2?)j 4 2relz 4 2y 4 22k vektormező integrálja a 0 £ r £ 1, 0 £ y £ 1, 0 £ 2 £ 1 egységkocka felületén kifelé 2. Határozzuk meg az ur y,2) — 2214. (—2y 4 2rz — y2)j 4 (2? 422 4 2jk vektormező (a.0.0). B — (0,a.0) és €— (0.0.a). a — 0. 3. Integráljuk az u(z, y. az irányban), ahol A 4. Integráljuk a víz, y. forgásiránnyal. "További gyakorló feladatok 6. Mennyi az ulz, y.2) — (zy 452214 2j 4 (z2 — vP)k vektormező integrálja annak a tetra- (0,—3.0), (4.3.3)7 4 zyk vektormező integrálját az x 2 0.y 2 02 2 7. Mennyi az ulr, y.2) — (ry 4 y2)i 4. (2? — y2)j 4 (2zy 4. 2?)k vektormező integrálja az LET] hatelbaj a9 Jt34-Dk hatel2j (6-Y14(3-1j4k hate[i (4-Yk hateBAj görbe ! € 8. Határozzuk meg az ulr. y.2) — (1 — 24214 (— gy2lj 4 (ye? — a22)k vektormező integrálját az r() — (F 2 tji 4 VTT 3E görbe ( € [-2,2] intervallumnak megfelelő szakaszán. (0, a) darabján? 9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol / skalármező, u vektormező, . peremes irányított felület. [docuuta9 — [. fu-dc— [otr u] 45 10 Bizonyítsuk be, hogy egy T síkidom területét h fgot te sörbével határolt asztroid területét.
9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol f skalármező, u vektormező, S peremes irányított felület: Z S(f rot u) · dA = Z ∂S fu · dr − Z S(grad f × u) · dA Megoldás. A Stokes-tétel szerint Z ∂S fu · dr = Z S rot(fu) · dA teljesül. A jobb oldalon az integrandust Leibniz-szabály szerint lehet kifejteni: rot(fu) = grad f × u + f rot u. Ebből átrendezéssel adódik az állítás. A felhasznált Leibniz-szabály a komponensek kiszámolásával ellenőrizhető, pl. rot(fu)z = ∂(fuy) ∂x (fuy) ∂x −∂(fux) ∂y ∂y = ∂f ∂ ∂f ∂xuy + f ∂uy ∂x ∂uy ∂x −∂f ∂y ∂f ∂y ux −f ∂ux ∂y ∂xuy + f ∂x ∂y ux f ∂y = (grad f × u)z + f(rot u)z, stb. 10. Bizonyítsuk be, hogy egy T síkidom területét A = I ∂T xj · dr = − I ∂T yi · dr módon is számíthatjuk és ennek segítségével határozzuk meg az r(t) = cos3 ti + sin3 tj görbével határolt asztroid területét. Megoldás. A v(x, y, z) = xj és a w(x, y, z) = −yi vektormezők rotációja rot v(x, y, z) = rot w(x, y, z) = k, így a Green-tétel szerint valóban a területet kapjuk. ˙r(t) = −3 cos2 t sin ti + 3 sin2 t cos tj miatt A = Z 2π cos3 t(3 sin2 t cos t)dt = 3 Z 2π cos4 t sin2 tdt 0Z 2π 1 = 3 1 16 + cos 2t 32 os 2t 32 −cos 4t 16 os 4t 16 −cos 6t 32 32 dt = 3π 8	9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol f skalármező, u vektormező, _S peremes irányított felület:_ � � � _S[(][f][ rot][ u][)][ ·][ d][A][ =]_ _∂S_ _[f]_ [u][ ·][ d][r][ −] _S[(grad][ f][ ×][ u][)][ ·][ d][A]_ _Megoldás. A Stokes-tétel szerint_ � � _∂S_ _[f]_ [u][ ·][ d][r][ =] _S_ [rot(][f] [u][)][ ·][ d][A] teljesül. A jobb oldalon az integrandust Leibniz-szabály szerint lehet kifejteni: rot(f u) = grad f × u + f rot u. Ebből átrendezéssel adódik az állítás. A felhasznált Leibniz-szabály a komponensek kiszámolásával ellenőrizhető, pl. rot(f u)z = _[∂][(][fu][y][)]_ _−_ _[∂][(][fu][x][)]_ _∂x_ _∂y_ = _[∂f]_ _∂x_ _[u][y][ +][ f ∂u]∂x[y]_ _[−]_ _[∂f]∂y [u][x][ −]_ _[f ∂u]∂y[x]_ = (grad f × u)z + f (rot u)z, stb. 10. Bizonyítsuk be, hogy egy T síkidom területét � � _A =_ _∂T_ _[x][j][ ·][ d][r][ =][ −]_ _∂T_ _[y][i][ ·][ d][r]_ módon is számíthatjuk és ennek segítségével határozzuk meg az r(t) = cos[3] _ti + sin[3]_ _tj_ görbével határolt asztroid területét. _Megoldás. A v(x, y, z) = xj és a w(x, y, z) = −yi vektormezők rotációja rot v(x, y, z) =_ rot w(x, y, z) = k, így a Green-tétel szerint valóban a területet kapjuk. r˙(t) = −3 cos[2] _t sin ti + 3 sin[2]_ _t cos tj miatt_ � 2π � 2π _A =_ cos[3] _t(3 sin[2]_ _t cos t)dt = 3_ cos[4] _t sin[2]_ _tdt_ 0 0 � 2π � 1 � = 3 _−_ [cos 4][t] _−_ [cos 6][t] _dt = [3][π]_ 0 16 [+ cos 2]32 _[t]_ 16 32 8 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Bizonyítsuk be az alábbi parciális integrálási formulát, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> skalármező,</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező,</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> peremes irányított felület:</span></p> <p style="top:90.4pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:114.6pt;left:112.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> rot</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:90.4pt;left:201.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:114.6pt;left:207.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂S</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:90.4pt;left:270.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:114.6pt;left:275.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(grad</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup></p> <p style="top:132.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A Stokes-tétel szerint</span></p> <p style="top:149.5pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:173.7pt;left:112.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂S</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:149.5pt;left:175.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:173.7pt;left:181.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup></p> <p style="top:192.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">teljesül. A jobb oldalon az integrandust Leibniz-szabály szerint lehet kifejteni:</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> rot(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:206.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">grad</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ebből átrendezéssel adódik az állítás. A felhasznált Leibniz-szabály</span></p> <p style="top:221.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a komponensek kiszámolásával ellenőrizhető, pl.</span></p> <p style="top:253.7pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">fu</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:261.9pt;left:179.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:253.7pt;left:207.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">fu</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:261.9pt;left:231.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:284.6pt;left:155.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂f</span></i></sup></p> <p style="top:292.8pt;left:169.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f ∂u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup></p> <p style="top:292.8pt;left:220.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂f</span></i></sup></p> <p style="top:292.8pt;left:254.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f ∂u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:292.8pt;left:306.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:309.2pt;left:155.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (grad</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:334.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">stb.</span></p> <p style="top:354.2pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Bizonyítsuk be, hogy egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> T</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkidom területét</span></p> <p style="top:383.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:371.0pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">I</span></p> <p style="top:395.3pt;left:136.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂T</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:371.0pt;left:208.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">I</span></p> <p style="top:395.3pt;left:213.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂T</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:414.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">módon is számíthatjuk és ennek segítségével határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:429.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbével határolt asztroid területét.</span></p> <p style="top:448.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezők rotációja</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:462.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, így a Green-tétel szerint valóban a területet kapjuk.</span></p> <p style="top:477.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">˙</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3 sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> miatt</span></p> <p style="top:509.3pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:496.8pt;left:133.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:521.0pt;left:138.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:509.3pt;left:154.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(3 sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dt</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span></p> <p style="top:496.8pt;left:281.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:521.0pt;left:287.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:509.3pt;left:303.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">tdt</span></i></p> <p style="top:537.9pt;left:120.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 3</span></p> <p style="top:525.4pt;left:140.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:549.6pt;left:146.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:524.9pt;left:162.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></p> <p style="top:546.1pt;left:171.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ cos 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:546.1pt;left:207.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span></p> <p style="top:537.9pt;left:231.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:546.1pt;left:252.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span></p> <p style="top:537.9pt;left:276.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 6</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:546.1pt;left:297.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span></p> <p style="top:524.9pt;left:318.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:537.9pt;left:327.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dt</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></sup></p> <p style="top:546.1pt;left:358.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_265.png	10. ítsuk be az alábbi parciális integrálási formulát, ahol / skalármező, u vektormező, " [docuutk — [ga-de- [dosaj o-x .Megoldás. A Stökes-tétel szerint [ayacae- [tx teljesül. A jobb oldalon az integrandust Leibniz-szabály szerint lehet kifejteni: rot(fu) — grad / x u 4. frotu. Ebből átrendezéssel adódik az állítás. A felhasznált Leibuiz-szabály a komponensek kiszámolásával ellenőrizhető, pl. ölfuj) . ölfuz) rotlfa9e s — ö 01 , , j04, O, te t — (grad f x u). 4 f(rot ).. Bizonyítsuk be, hogy egy T síkidom területét A fzbdefeote ! görbével határolt asztroid terüle "Megoldás. A víz,y. 7j és a W(x. y.2) — —yi vektormezők rotációja rot v(z, v. rot Ww(z. y.2) — k. így a Green-tétel szerint valóban a területet kapjuk. é(t) — 300" tsinti 4. 3sin? t cosj miatt costtsin? édt 1 E ] 16
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 6. feladatsor: Integrálátalakító tételek (megoldás) 1. Mennyi az u(x, y, z) = x(x −2xy + 2yz2)i −y(2x2 + 4xyz + yz2)j + 2xz(x + 2y + 2yz)k vektormező integrálja a 0 ≤x ≤1, 0 ≤y ≤1, 0 ≤z ≤1 egységkocka felületén kifelé mutató irányítás mellett? Megoldás. Alkalmazzuk a Gauss-Osztrogradszkij-tételt: div u = (2x −4xy + 2yz2) −(2x2 + 8xyz + 2yz2) + (2x2 + 4xy + 8xyz) = 2x Z 1 Z 1 Z 1 0 2xdxdydz = 1 Z 1 ZZ ∂V u · dA = ZZ ZZZ V div u dV = ZZZ 2. Határozzuk meg az u(x, y, z) = −xzi + (−xy + 2xz −yz)j + (x2 + xz + z2)k vektormező integrálját azon az irányított felületen, amely az x2 + y2 ≤1, 0 ≤z ≤10 −x egyenlőtlenségek által meghatározott tartomány kifelé irányított felületéből a z = 0 síkba eső alapkör elhagyásával keletkezik. 2. Határozzuk meg az u(x, y, z) = −xzi + (−xy + 2xz −yz)j + (x2 + xz + z2)k vektormező Megoldás. Ha hozzávennénk az elhagyott körlapot a felülethez, akkor zárt felületet kapunk, amin az integrált a Gauss-Osztrogradszkij-tétellel is számolhatjuk. div u(x, y, z) = ∂(−xz) ∂ −xz) ∂x + ∂(−xy + 2xz −yz) ∂y + 2xz −yz) ∂y + ∂(x2 + xz + z2) ∂z + xz + z ) ∂z = −z−x−z+x+2z = 0. Ha a megadott tartományt V jelöli, az elhagyott (kifelé irányított) körlapot pedig S, akkor 0 = Z V div u dV = Z ∂V u · dA = Z S u · dA + Z ∂V \S u · dA, tehát a keresett integrál az S-en vett integrál ellentettje. (Úgy is lehetne érvelni, hogy div u = 0 miatt az integrál csak a határoló irányított görbétől függ, tehát ugyanaz, mint az integrál S-en megfordított irányítással.) Paraméterezzük az S körlapot r(r, φ) = r cos φi + r sin φj módon, (r, φ) ∈[0, 1] × [0, 2π]. A normálvektor ∂r ∂r × ∂r ∂φ = (cos φi + sin φj) × (−r sin φi + r cos φj) = rk, de ez felfelé mutat, tehát a megfelelő irányításhoz az ellentettjét kell venni. A vektormező a felületen u(r(r, φ)) = −r2 cos φ sin φj + r2 cos2 φk, így a keresett integrál: I = − = − Z S u · dA S Z 2π Z 1 Z 1 0 (−r2 cos φ sin φj + r2 cos2 φk) · (−rk) dr dφ 0 Z 2π Z 1 cos2 φ dφ Z 1 4 0 r3 dr = π π 4 . 3. Integráljuk az u(x, y, z) = y2i + z2j + x2k vektormezőt az ABC háromszögvonalon (ebben az irányban), ahol A = (a, 0, 0), B = (0, a, 0) és C = (0, 0, a), a > 0.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 6. feladatsor: Integrálátalakító tételek (megoldás) 1. Mennyi az u(x, y, z) = x(x − 2xy + 2yz[2])i − _y(2x[2]_ + 4xyz + yz[2])j + 2xz(x + 2y + 2yz)k vektormező integrálja a 0 ≤ _x ≤_ 1, 0 ≤ _y ≤_ 1, 0 ≤ _z ≤_ 1 egységkocka felületén kifelé mutató irányítás mellett? _Megoldás. Alkalmazzuk a Gauss-Osztrogradszkij-tételt:_ div u = (2x − 4xy + 2yz[2]) − (2x[2] + 8xyz + 2yz[2]) + (2x[2] + 4xy + 8xyz) = 2x �� 1 � 1 � 1 _∂V_ [u][ ·][ d][A][ =] _V_ [div][ u][ d][V][ =] 0 0 0 [2][xdxdydz][ = 1] 2. Határozzuk meg az u(x, y, z) = −xzi + (−xy + 2xz − _yz)j + (x[2]_ + xz + z[2])k vektormező integrálját azon az irányított felületen, amely az x[2] + y[2] _≤_ 1, 0 ≤ _z ≤_ 10 − _x egyenlőtlen-_ ségek által meghatározott tartomány kifelé irányított felületéből a z = 0 síkba eső alapkör elhagyásával keletkezik. _Megoldás. Ha hozzávennénk az elhagyott körlapot a felülethez, akkor zárt felületet kapunk,_ amin az integrált a Gauss-Osztrogradszkij-tétellel is számolhatjuk. div u(x, y, z) = _[∂][(][−][xz][)]_ + _[∂][(][−][xy][ + 2][xz][ −]_ _[yz][)]_ + _[∂][(][x][2][ +][ xz][ +][ z][2][)]_ = −z _−x−z_ +x+2z = 0. _∂x_ _∂y_ _∂z_ Ha a megadott tartományt V jelöli, az elhagyott (kifelé irányított) körlapot pedig S, akkor � � � � 0 = _V_ [div][ u][ d][V][ =] _∂V_ [u][ ·][ d][A][ =] _S_ [u][ ·][ d][A][ +] _∂V \S_ [u][ ·][ d][A][,] tehát a keresett integrál az S-en vett integrál ellentettje. (Úgy is lehetne érvelni, hogy div u = 0 miatt az integrál csak a határoló irányított görbétől függ, tehát ugyanaz, mint az integrál S-en megfordított irányítással.) Paraméterezzük az S körlapot r(r, φ) = r cos φi + r sin φj módon, (r, φ) ∈ [0, 1] × [0, 2π]. A normálvektor _∂r_ _∂r_ _[×][ ∂]∂φ[r]_ [= (cos][ φ][i][ + sin][ φ][j][)][ ×][ (][−][r][ sin][ φ][i][ +][ r][ cos][ φ][j][) =][ r][k][,] de ez felfelé mutat, tehát a megfelelő irányításhoz az ellentettjét kell venni. A vektormező a felületen u(r(r, φ)) = −r[2] cos φ sin φj + r[2] cos[2] _φk,_ így a keresett integrál: � _I = −_ _S_ [u][ ·][ d][A] � 2π � 1 = − 0 0 [(][−][r][2][ cos][ φ][ sin][ φ][j][ +][ r][2][ cos][2][ φ][k][)][ ·][ (][−][r][k][) d][r][ d][φ] � 2π � 1 = cos[2] _φ dφ_ 0 0 _[r][3][ d][r][ =][ π]4_ _[.]_ 3. Integráljuk az u(x, y, z) = y[2]i + z[2]j + x[2]k vektormezőt az ABC háromszögvonalon (ebben az irányban), ahol A = (a, 0, 0), B = (0, a, 0) és C = (0, 0, a), a > 0. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:91.4pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">6. feladatsor: Integrálátalakító tételek (megoldás)</span></b></p> <p style="top:131.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:146.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektormező integrálja a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egységkocka felületén kifelé</span></p> <p style="top:160.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mutató irányítás mellett?</span></p> <p style="top:179.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Alkalmazzuk a Gauss-Osztrogradszkij-tételt:</span></p> <p style="top:205.1pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:223.6pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">ZZ</span></p> <p style="top:247.8pt;left:118.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂V</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:223.6pt;left:180.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">ZZZ</span></p> <p style="top:247.8pt;left:197.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">V</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">V</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:223.6pt;left:266.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:247.8pt;left:271.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:223.6pt;left:283.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:247.8pt;left:288.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:223.6pt;left:299.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:247.8pt;left:305.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xdxdydz</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup></p> <p style="top:261.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező</span></p> <p style="top:275.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">integrálját azon az irányított felületen, amely az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenlőtlen-</span></p> <p style="top:290.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ségek által meghatározott tartomány kifelé irányított felületéből a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkba eső alapkör</span></p> <p style="top:304.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elhagyásával keletkezik.</span></p> <p style="top:323.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ha hozzávennénk az elhagyott körlapot a felülethez, akkor zárt felületet kapunk,</span></p> <p style="top:337.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amin az integrált a Gauss-Osztrogradszkij-tétellel is számolhatjuk.</span></p> <p style="top:369.8pt;left:85.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:378.0pt;left:178.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:369.8pt;left:206.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:378.0pt;left:259.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:369.8pt;left:316.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:378.0pt;left:361.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂z</span></i></p> <p style="top:369.8pt;left:412.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:401.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha a megadott tartományt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> V</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelöli, az elhagyott (kifelé irányított) körlapot pedig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor</span></p> <p style="top:430.2pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span></p> <p style="top:417.7pt;left:128.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:441.9pt;left:133.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">V</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">V</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:417.7pt;left:202.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:441.9pt;left:207.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂V</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:417.7pt;left:270.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:441.9pt;left:275.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:417.7pt;left:331.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:441.9pt;left:336.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂V</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> \</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:463.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a keresett integrál az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-en vett integrál ellentettje. (Úgy is lehetne érvelni, hogy</span></p> <p style="top:478.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> miatt az integrál csak a határoló irányított görbétől függ, tehát ugyanaz, mint</span></p> <p style="top:492.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az integrál</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-en megfordított irányítással.)</span></p> <p style="top:507.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Paraméterezzük az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> körlapot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> módon,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A</span></p> <p style="top:521.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">normálvektor</span></p> <p style="top:541.7pt;left:107.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:558.0pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂r</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:558.0pt;left:137.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:580.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">de ez felfelé mutat, tehát a megfelelő irányításhoz az ellentettjét kell venni. A vektormező</span></p> <p style="top:594.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a felületen</span></p> <p style="top:620.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r, φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:645.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">így a keresett integrál:</span></p> <p style="top:674.0pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:661.5pt;left:141.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:685.8pt;left:147.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup></p> <p style="top:702.6pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:690.1pt;left:141.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:714.4pt;left:147.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:690.1pt;left:163.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:714.4pt;left:168.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> φ</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i></sup></p> <p style="top:731.2pt;left:117.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:718.7pt;left:130.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:743.0pt;left:135.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:731.2pt;left:152.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">φ</span></i></p> <p style="top:718.7pt;left:198.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:743.0pt;left:204.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π</span></i></sup></p> <p style="top:739.4pt;left:257.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:761.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Integráljuk az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ABC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> háromszögvonalon (ebben</span></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az irányban), ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, a,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> </div>	page_266.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 6. feladatsor: Integrálátalakító tételek (megoldás) 1. Mennyi az ülr,y.2) — 2(r — 2xy 4 2y22)i — y(2r? 4 deye 4 y2?)j 4 2relz 4 2y 4 22k vektormező integrálja a 0 £ r £ 1, 0 £ y £ 1, 0 £ 2 £ 1 egységkocka felületén kifelé .Megoldás. Alkalmazzuk a Gauss-Osztrogradszkij-tételt: dívu — (2r— dry 4 242) — (22? 4 Bzyz 4 2y2?) 4 22 4 dry - 8r s 48 - [ffjaseuav — [ [9 [/ovtedy 2. Határozzuk meg az ur y, 2214 (—zy 4. 2rz — y2)j 4 ( 422 z?jk vektormező tegrálját azon az irányított felületen, amely az 2 4- ? £ 1, 0 £ : £ 10 — 2 egyenlőtlet égek által meghatározott tartomány kifelé irányított felületéből a : — 0 síkba eső alapkör "Megoldás. Ha hozzávennénk az elhagyott körlapot a felülethez, akkor 2árt felületet kapunk, Öl-zz) , Ol-zy 4-2x2 — y2) , T 4 m E [7 ? ; dívudV m. dA, Paraméterezzük az § közlapot rr. ó) — reosói 4-rsin éj módon, (r.6) € [0.1] x [/.271. A ör , 0r 2E , 3L — (cosói 4-sin ój) x (-rsindi 4 reoséj) — rk. F ] ulrlr, )) — —r cosósin éj 4 17 cos? ók így a keresett integról. [EE 2 [/[/oíAuusósínéj 41 co ékj : (-rkjdrdó [őcszódó ar 3 az irányban), ahol A (a.0.0). § — (0,a.0) és C-— (0.0.a), a — 0. 1
Megoldás. Alkalmazzuk a Stokes-tételt: ∂uz ∂ux ∂uy rot u = ∂uz ∂y −∂uy ∂z ∂z i + ∂ux ∂z −∂uz ∂x ∂x j + ∂uy ∂x −∂ux ∂y ∂y k = −2zi −2xj −2yk A háromszög paraméterezése: r(u, v) = ai + ua(j −i) + va(k −i) Z ∂S u · dr = Z S rot u · dA Z 1 Z 1−u = a3 = a3 0 Z 1 0 Z 1−u (−2vi −2(1 −u −v)j −2uk) · (j −i) × (k −i) dv du (−2) dv du = −a3 4. Integráljuk a v(x, y, z) = −x2yi+xy2j vektormezőt az x2 +y2 = a2 egyenletű körön pozitív forgásiránnyal. Megoldás. Alkalmazzuk a Green-tételt, rot v(x, y, z) = (x2 + y2)k, tehát Z a Z 2π r2rdϕdr = 2πa4 4 ZZ ⃝rot v · dA = a4 4 = π 2 ZZ I ∂⃝v · dr = π 2 a4 További gyakorló feladatok 5. Számítsuk ki az u(x, y, z) = yzi + xzj + xyk vektormező integrálját az x ≥0, y ≥0, z ≥ 0, x + y + z ≤3 tetraéder felületén kifelé mutató irányítás mellett. Megoldás. div u = 0, ezért u zárt felületen vett integrálja 0. 6. Mennyi az u(x, y, z) = (xy + 5z2)i + y2j + (xz −y2)k vektormező integrálja annak a tetra- édernek a felületén kifelé mutató irányítás mellett, amelynek csúcsai (1, −2, 3), (0, −1, 1), (0, −3, 0), (4, 3, 3)? Megoldás. Zárt felületen kell integrálni, tehát használható a Gauss-Osztrogradszkij-tétel. Eszerint a kérdéses integrál megegyezik div u integráljával a T “tömör” tetraéderen. Egy kényelmes paraméterezéshez tekintsük az (1, −2, 3) csúcsból kiinduló a, b, c élvektorokat: a = −i + j −2k b = −i −j −3k c = 3i + 5j. A paraméterezés legyen r(u, v, w) = (i −2j + 3k) + ua + vb + wc, ahol 0 ≤u ≤1, 0 ≤v ≤1 −u, 0 ≤w ≤1 −u −v. Ekkor a Jacobi-determináns \|abc\| = 20. A vektormező divergenciája: div u(x, y, z) = x + 3y div u(r(u, v, w)) = −5 + 2u −4v + 18w, tehát Z ∂T u · dA = Z T div u(x, y, z) · dx dy dz T Z 1 Z 1−u Z 1−u−v 20 (−5 + 2u −4v + 18w) dw dv du 0 Z 1 0 Z 1 0 Z 1−u 80 −220u + 140u2 −340v + 400uv + 260v2 dv du −10 3 −20u + 50u2 −80u3 3 10 3 −20u + 50u2 −80u3 3 du = −10 3 . = −10 3	_Megoldás. Alkalmazzuk a Stokes-tételt:_ � i + �∂ux _∂z_ _[−]_ _[∂u]∂x[z]_ � j + �∂uy _∂x_ _[−]_ _[∂u]∂y[x]_ � k = −2zi − 2xj − 2yk rot u = �∂uz _∂y_ _[−]_ _[∂u]∂z[y]_ A háromszög paraméterezése: r(u, v) = ai + ua(j − i) + va(k − i) � � _∂S_ [u][ ·][ d][r][ =] _S_ [rot][ u][ ·][ d][A] � 1 � 1−u = a[3] (−2vi − 2(1 − _u −_ _v)j −_ 2uk) · (j − i) × (k − i) dv du 0 0 � 1 � 1−u = a[3] (−2) dv du = −a[3] 0 0 4. Integráljuk a v(x, y, z) = −x[2]yi + _xy[2]j vektormezőt az x[2]_ + _y[2]_ = a[2] egyenletű körön pozitív forgásiránnyal. _Megoldás. Alkalmazzuk a Green-tételt, rot v(x, y, z) = (x[2]_ + y[2])k, tehát � �� _a_ � 2π _r[2]rdϕdr = 2π_ _[a][4]_ _∂⃝_ [v][ ·][ d][r][ =] _⃝_ [rot][ v][ ·][ d][A][ =] 0 0 4 [=][ π]2 _[a][4]_ ## További gyakorló feladatok 5. Számítsuk ki az u(x, y, z) = yzi + xzj + xyk vektormező integrálját az x ≥ 0, y ≥ 0, z ≥ 0, x + y + z ≤ 3 tetraéder felületén kifelé mutató irányítás mellett. _Megoldás. div u = 0, ezért u zárt felületen vett integrálja 0._ 6. Mennyi az u(x, y, z) = (xy + 5z[2])i + y[2]j + (xz − _y[2])k vektormező integrálja annak a tetra-_ édernek a felületén kifelé mutató irányítás mellett, amelynek csúcsai (1, −2, 3), (0, −1, 1), (0, −3, 0), (4, 3, 3)? _Megoldás. Zárt felületen kell integrálni, tehát használható a Gauss-Osztrogradszkij-tétel._ Eszerint a kérdéses integrál megegyezik div u integráljával a T “tömör” tetraéderen. Egy kényelmes paraméterezéshez tekintsük az (1, −2, 3) csúcsból kiinduló a, b, c élvektorokat: a = −i + j − 2k b = −i − j − 3k c = 3i + 5j. A paraméterezés legyen r(u, v, w) = (i − 2j + 3k) + ua + vb + wc, ahol 0 ≤ _u ≤_ 1, 0 ≤ _v ≤_ 1 − _u, 0 ≤_ _w ≤_ 1 − _u −_ _v. Ekkor a Jacobi-determináns \|abc\| = 20._ A vektormező divergenciája: div u(x, y, z) = x + 3y div u(r(u, v, w)) = −5 + 2u − 4v + 18w, tehát � � _∂T_ [u][ ·][ d][A][ =] _T_ [div][ u][(][x, y, z][)][ ·][ d][x][ d][y][ d][z] � 1 � 1−u � 1−u−v = 20 (−5 + 2u − 4v + 18w) dw dv du 0 0 0 = � 1 � 1−u �80 − 220u + 140u[2] _−_ 340v + 400uv + 260v[2][�] dv du 0 0 1 � � � = _−[10]_ du 0 3 _[−]_ [20][u][ + 50][u][2][ −] [80]3[u][3] = −[10] 3 _[.]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Alkalmazzuk a Stokes-tételt:</span></p> <p style="top:87.1pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:71.1pt;left:146.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></p> <p style="top:95.3pt;left:158.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup></p> <p style="top:95.3pt;left:193.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂z</span></i></p> <p style="top:71.1pt;left:210.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:87.1pt;left:220.5pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:71.1pt;left:238.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:95.3pt;left:250.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup></p> <p style="top:95.3pt;left:285.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:71.1pt;left:303.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:87.1pt;left:312.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:71.1pt;left:331.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:95.3pt;left:343.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:95.3pt;left:378.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:71.1pt;left:395.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:87.1pt;left:405.8pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:115.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A háromszög paraméterezése:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ua</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> va</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:128.1pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:152.3pt;left:114.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂S</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:128.1pt;left:170.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:152.3pt;left:176.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup></p> <p style="top:169.2pt;left:158.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:156.7pt;left:183.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:180.9pt;left:188.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:156.7pt;left:200.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></p> <p style="top:180.9pt;left:205.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:169.2pt;left:226.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:197.8pt;left:158.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:185.3pt;left:183.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:209.5pt;left:188.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:185.3pt;left:200.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></p> <p style="top:209.5pt;left:205.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:197.8pt;left:226.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:223.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Integráljuk a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormezőt az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletű körön pozitív</span></p> <p style="top:238.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">forgásiránnyal.</span></p> <p style="top:255.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Alkalmazzuk a Green-tételt,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:271.0pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">I</span></p> <p style="top:295.3pt;left:112.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">⃝</span></i><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:271.0pt;left:171.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">ZZ</span></p> <p style="top:295.3pt;left:182.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">⃝</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:271.0pt;left:258.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> a</span></i></p> <p style="top:295.3pt;left:263.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:271.0pt;left:275.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">π</span></i></p> <p style="top:295.3pt;left:280.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:283.5pt;left:296.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">r</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">rdϕdr</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:291.7pt;left:370.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π</span></i></sup></p> <p style="top:291.7pt;left:397.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:318.5pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:342.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Számítsuk ki az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálját az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i></p> <p style="top:357.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tetraéder felületén kifelé mutató irányítás mellett.</span></p> <p style="top:374.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ezért</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> zárt felületen vett integrálja</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:391.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja annak a tetra-</span></p> <p style="top:405.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">édernek a felületén kifelé mutató irányítás mellett, amelynek csúcsai</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:420.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:437.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Zárt felületen kell integrálni, tehát használható a Gauss-Osztrogradszkij-tétel.</span></p> <p style="top:451.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Eszerint a kérdéses integrál megegyezik</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> integráljával a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> T</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> “tömör” tetraéderen. Egy</span></p> <p style="top:466.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kényelmes paraméterezéshez tekintsük az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> csúcsból kiinduló</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> b</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> élvektorokat:</span></p> <p style="top:487.6pt;left:107.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:505.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:522.4pt;left:107.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 5</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:543.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A paraméterezés legyen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v, w</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> w</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:558.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ekkor a Jacobi-determináns</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">abc</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 20</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:572.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A vektormező divergenciája:</span></p> <p style="top:593.7pt;left:123.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:611.2pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v, w</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 18</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w,</span></i></p> <p style="top:632.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> Z</span></p> <p style="top:665.9pt;left:114.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂T</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:641.7pt;left:176.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:665.9pt;left:181.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">T</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">div</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup></p> <p style="top:682.8pt;left:163.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:670.3pt;left:176.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:694.5pt;left:181.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:670.3pt;left:192.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></p> <p style="top:694.5pt;left:198.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:670.3pt;left:220.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">v</span></i></p> <p style="top:694.5pt;left:226.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:682.8pt;left:260.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">20 (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 18</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:711.4pt;left:163.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:698.9pt;left:176.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:723.1pt;left:181.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:698.9pt;left:192.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></p> <p style="top:723.1pt;left:198.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:701.4pt;left:220.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:711.4pt;left:226.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">80</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">220</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 140</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">340</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 400</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">uv</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 260</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></sup></p> <p style="top:711.4pt;left:454.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:741.9pt;left:163.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:729.4pt;left:176.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:753.6pt;left:181.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:725.9pt;left:192.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span></p> <p style="top:741.9pt;left:200.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span></sup></p> <p style="top:750.1pt;left:214.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">20</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 50</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">80</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:750.1pt;left:318.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:725.9pt;left:334.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:741.9pt;left:344.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:773.4pt;left:163.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span></sup></p> <p style="top:781.6pt;left:189.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_267.png	A háromszög paraméterezése: r(u.v) [ot [1019-4A " [[/C2A-20-4-9)—249-(-1) x (k-gydede 4. Integráljuk a víz, y. forgásiránnyal. . Megoldás. Alkalmazzuk a Grec fovten [[dovcAAof] [7 érdete a é "További gyakorló feladatok .Megoldás, div u — 0. ezért u zárt felületen vett intesrálja 0. 6. Mennyi az ulz, y.2) — (zy 452114 Éj 4 (22 — vP)k vektormező integrálja annak a tetra- (0,—3.0), (4.3.3)? árt felüle v zyéj vektormezőt az £ a? egyenletű körön pozíté ételt, rot v. y. 2 4 y)k, tehát 4 zyk vektormező integrálját az 7 2 0. y 2 0. . Megoldás. m kell integrálni, tehát használható a Gauss-Osztrogradszkij-tétel. Eszerint a kérdéses inte relmes paraméterezéshez tekintsük az (1, a-—-itj-2k 3k gyezik dívu integráljával a T "tömör" tetraéderen. Egy 3) csúcsból kiinduló a, b, c élvektorokat: A param legyen ríu,z,1w) — (1— 2j 4-3k) 4 ua 4-vb 4 awc, ahol 0 £ a £ 1. OsvE1-—u.0£0£1—u-tr. Ekkor a Jacobi-determináns [abel A vektormező dívergenciája: dívu(z,y,2) 2r 3 dívu(r(u, v.19) — —5 4-24 — 4r 180 4994 [/ddvAtr 2) drdyds " /D'Á"" /!.HH 20(—5 4- 24 — 44 4 ISe)duwdedu -[ 500 - Zs 10 3
(Lehetett volna közvetlenül is számolni az integrált, de az laponként egy kétváltozós integrál kiszámításával jár, így egyszerűbb.) 7. Mennyi az u(x, y, z) = (xy + yz)i + (x2 −yz)j + (2xy + z2)k vektormező integrálja az ti + t2j ha t ∈[0, 1] i + j + (t −1)k ha t ∈[1, 2] (3 −t)i + (3 −t)2j + k ha t ∈[2, 3] (4 −t)k ha t ∈[3, 4] r(t) = görbe t ∈[0, 4] darabján? Megoldás. A megadott görbe zárt, tehát használhatjuk a Stokes-tételt. Ehhez egy S irányított felületdarabot kell választani, aminek a pereme éppen az adott görbe. Egy kényelmes választás: r(u, v) = ui+u2j+vk (egy parabolikus henger darabja), ahol (u, v) ∈[0, 1]×[0, 1]. A parciális deriváltak vektoriális szorzata ∂r ∂u × ∂r ∂v = (i + 2uj) × k = 2ui −j, ez a vektor az irányítással ellentétes irányba mutat. A felületen az u vektormező rotációját kell integrálni: rot u(x, y, z) = (2x + y)i + (−y)j + (x −z)k rot u(r(u, v)) = (2u + u2)i −u2j + (u −v)k, tehát Z ∂S u · dr = Z S rot u · dA Z 1 Z 1 ∂r ∂r ∂u × ∂r ∂v = − = − Z 0 u(r(u, v)) ∂v du dv Z 1 Z 1 Z 1 6 0 (5u2 + 2u3) du dv = −13 13 6 . 8. Határozzuk meg az u(x, y, z) = (xy2 −y2z + x2)i + (x2y −xyz)j + (yz2 −x2z)k vektormező g ( , y, ) integrálját az r(t) = (t3 + t)i + √ g ( , y, ) ( y y + ) + ( y y )j + (y ) integrálját az r(t) = (t3 + t)i + 4 + 3t2 −t4j görbe t ∈[−2, 2] intervallumnak megfelelő szakaszán. Megoldás. Ha a görbéhez hozzáfűzzük a két végpontot összekötő szakaszt, akkor zárt görbét kapunk, tehát alkalmazható a Stokes-tétel. rot u(x, y, z) = (xy + z2)i + (−y2 + 2xz)j + yzk A kapott zárt görbe (és így az általa határolt síkidom) a z = 0 síkban van, ott a rotáció z irányú komponense 0, tehát a felületi integrál 0. Eszerint a kérdéses integrál ugyanaz, mintha a két végpontot összekötő egyenesszakaszon integrálnánk. r(t) = ti (t ∈[−10, 10]) ennek egy paraméterezése, tehát u · dr = Z 10 Z −10 u(t, 0, 0) · i dt Z 10 Z −10 ux(t, 0, 0) dt "t3 #10 Z 10 −10 t2 dt = Z 10 = 2000 3 −10 3	(Lehetett volna közvetlenül is számolni az integrált, de az laponként egy kétváltozós integrál kiszámításával jár, így egyszerűbb.) 7. Mennyi az u(x, y, z) = (xy + yz)i + (x[2] _−_ _yz)j + (2xy + z[2])k vektormező integrálja az_ r(t) = ti + t[2]j ha t ∈ [0, 1] i + j + (t − 1)k ha t ∈ [1, 2] (3 − _t)i + (3 −_ _t)[2]j + k_ ha t ∈ [2, 3] (4 − _t)k_ ha t ∈ [3, 4] görbe t ∈ [0, 4] darabján? _Megoldás. A megadott görbe zárt, tehát használhatjuk a Stokes-tételt. Ehhez egy S irányí-_ tott felületdarabot kell választani, aminek a pereme éppen az adott görbe. Egy kényelmes választás: r(u, v) = ui+u[2]j+vk (egy parabolikus henger darabja), ahol (u, v) ∈ [0, 1]×[0, 1]. A parciális deriváltak vektoriális szorzata _∂r_ _∂u_ _[×][ ∂]∂v[r]_ [= (][i][ + 2][u][j][)][ ×][ k][ = 2][u][i][ −] [j][,] ez a vektor az irányítással ellentétes irányba mutat. A felületen az u vektormező rotációját kell integrálni: rot u(x, y, z) = (2x + y)i + (−y)j + (x − _z)k_ rot u(r(u, v)) = (2u + u[2])i − _u[2]j + (u −_ _v)k,_ tehát � � _∂S_ [u][ ·][ d][r][ =] _S_ [rot][ u][ ·][ d][A] � 1 = − 0 � 1 = − 0 � 1 0 [u][(][r][(][u, v][))] � 1 0 [(5][u][2][ + 2][u][3][) d][u][ d][v][ =][ −][13]6 _[.]_ � _∂r_ _∂u_ _[×][ ∂]∂v[r]_ � du dv 8. Határozzuk meg az u(x, y, z) = (xy[2] _−_ _y[2]z + x[2])i + (x[2]y −_ _xyz)j + (yz[2]_ _−_ _x[2]z)k vektormező_ _√_ integrálját az r(t) = (t[3] + t)i + 4 + 3t[2] _−_ _t[4]j görbe t ∈_ [−2, 2] intervallumnak megfelelő szakaszán. _Megoldás. Ha a görbéhez hozzáfűzzük a két végpontot összekötő szakaszt, akkor zárt görbét_ kapunk, tehát alkalmazható a Stokes-tétel. rot u(x, y, z) = (xy + z[2])i + (−y[2] + 2xz)j + yzk A kapott zárt görbe (és így az általa határolt síkidom) a z = 0 síkban van, ott a rotáció _z irányú komponense 0, tehát a felületi integrál 0. Eszerint a kérdéses integrál ugyanaz,_ mintha a két végpontot összekötő egyenesszakaszon integrálnánk. r(t) = ti (t ∈ [−10, 10]) ennek egy paraméterezése, tehát � � 10 u · dr = _−10_ [u][(][t,][ 0][,][ 0)][ ·][ i][ d][t] � 10 = _−10_ _[u][x][(][t,][ 0][,][ 0) d][t]_ � 10 � _t3_ �10 = = [2000]. _−10_ _[t][2][ d][t][ =]_ 3 _−10_ 3 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(Lehetett volna közvetlenül is számolni az integrált, de az laponként egy kétváltozós integrál</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kiszámításával jár, így egyszerűbb.)</span></p> <p style="top:92.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Mennyi az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező integrálja az</span></p> <p style="top:144.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:110.0pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:119.0pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:121.9pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:124.9pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:127.9pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:130.9pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:133.9pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:151.8pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:154.8pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:157.8pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:160.8pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:163.8pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:166.8pt;left:141.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:118.7pt;left:149.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:118.7pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span></p> <p style="top:136.1pt;left:149.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:136.1pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span></p> <p style="top:153.4pt;left:149.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b></p> <p style="top:153.4pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3]</span></p> <p style="top:170.7pt;left:149.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:170.7pt;left:277.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4]</span></p> <p style="top:196.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">görbe</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> darabján?</span></p> <p style="top:215.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megadott görbe zárt, tehát használhatjuk a Stokes-tételt. Ehhez egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> irányí-</span></p> <p style="top:229.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tott felületdarabot kell választani, aminek a pereme éppen az adott görbe. Egy kényelmes</span></p> <p style="top:243.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">választás:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (egy parabolikus henger darabja), ahol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:258.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A parciális deriváltak vektoriális szorzata</span></p> <p style="top:280.3pt;left:108.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:296.6pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:296.6pt;left:138.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> k</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:316.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ez a vektor az irányítással ellentétes irányba mutat.</span></p> <p style="top:330.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A felületen az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező rotációját kell integrálni:</span></p> <p style="top:355.4pt;left:109.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:372.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:397.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span></p> <p style="top:410.6pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:434.8pt;left:114.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂S</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:410.6pt;left:170.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:434.8pt;left:176.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span></b></sup></p> <p style="top:453.5pt;left:158.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:441.1pt;left:182.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:465.3pt;left:187.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:441.1pt;left:198.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:465.3pt;left:204.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u, v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></sup></p> <p style="top:437.6pt;left:266.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></p> <p style="top:461.7pt;left:275.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b></sup></p> <p style="top:461.7pt;left:306.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂v</span></i></p> <p style="top:437.6pt;left:320.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:453.5pt;left:330.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></p> <p style="top:485.5pt;left:158.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:473.0pt;left:182.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:497.3pt;left:187.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:473.0pt;left:198.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></p> <p style="top:497.3pt;left:204.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(5</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13</span></sup></p> <p style="top:493.7pt;left:330.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:514.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yz</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektormező</span></p> <p style="top:529.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">integrálját az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:519.6pt;left:242.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:529.3pt;left:252.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> görbe</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> intervallumnak megfelelő</span></p> <p style="top:543.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szakaszán.</span></p> <p style="top:562.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ha a görbéhez hozzáfűzzük a két végpontot összekötő szakaszt, akkor zárt görbét</span></p> <p style="top:577.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kapunk, tehát alkalmazható a Stokes-tétel.</span></p> <p style="top:601.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rot</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> u</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y, z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xz</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yz</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">k</span></b></p> <p style="top:626.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kapott zárt görbe (és így az általa határolt síkidom) a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> síkban van, ott a rotáció</span></p> <p style="top:641.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> irányú komponense</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát a felületi integrál</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Eszerint a kérdéses integrál ugyanaz,</span></p> <p style="top:655.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mintha a két végpontot összekötő egyenesszakaszon integrálnánk.</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 10]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:669.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek egy paraméterezése, tehát</span></p> <p style="top:687.1pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:699.6pt;left:120.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">r</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:687.1pt;left:164.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 10</span></p> <p style="top:711.4pt;left:169.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">10</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></b><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">u</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> i</span></b></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:729.0pt;left:151.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:716.6pt;left:164.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 10</span></p> <p style="top:740.8pt;left:169.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup></p> <p style="top:762.3pt;left:151.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:749.9pt;left:164.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 10</span></p> <p style="top:774.1pt;left:169.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:746.4pt;left:224.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:770.5pt;left:233.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:746.4pt;left:241.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">10</span></p> <p style="top:778.0pt;left:247.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">10</span></p> <p style="top:762.3pt;left:266.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2000</span></sup></p> <p style="top:770.5pt;left:289.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:762.3pt;left:304.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_268.png	(Lehetett volna közvetlenül is számolni az integrált, de az laponként egy kétváltozós integrál kiszámításával jár, így egyszerűbb.) Mennyi az ulr, y.2) — (ry-4-y2)1 4 (22 — y2)j 4. (2ry 4 22k vektormező integrálja az LE hatelo1] d9-Athr] hatel.2] 6-914(3-9494k hatefa] (4-9k hate[34] görbe t € 0. 4j darabján? "Megoldás. A megadott görbe zárt, tehát használhatjuk a Stokcs-tételt. Ehhez egy $ irányí. tott felületdarabot kell választani, aminek a pereme éppen az adott görbe. Egy kényelmes választás: ríu, u) — ui4-t2j4-vk (egy parabolikus henger darabja), ahol (4, 2) € 0. 11210. 1 A parciális deríváltak vektoriális szorzata 0r ör ör e MYx ko -, ez a vektor az irányítással ellentétes irányba mutat. A felületen az u vektormező rotációját kell integrálni: rotu(z,y.2) — (2z 4 914 (—) 4 (r — 2k rot ulr(u, 1)) — (24 4 1998— 1234 (u — 0)k, tehát [dscdem [ot CCE [ (özé 4 2) dude Határozzuk meg az ülr.y.2) — (r — 124214 (r?y—zyaj 4 (y2? — a22)k vektormező integrálját az r(t) — (P 4-tji 4 VTFZEZT gőrbe ! € [-2,2] intervallumnak megfelelő szakaszán. . Megoldás. Ha a görbéhez hozzáfűzzük a két végpontot összekötő szakaszt, akkor zárt görbét kapunk, tehát alkalmazható a Stokcs-tétel. rot ú(r y.2) — (ry 44 ( 4 2724 2k A kapott 2árt görbe (és így az általa határolt síkidom) a : — 0 síkban van, ott a rotáció irányú komponense 0, tehát a felületi integrál 0. Eszerint a kérdéses integrál ugyanaz, ítha a két végpontot összekötő egyenesszakaszon integrálnánk. r() — fi (£ € [-10, 10) ennek egy paraméterezése, tehát -- fatt0.0 3t [dsAutoat
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 7. feladatsor: Szukcesszív approximáció, néhány egyenlettípus 1. Számoljuk ki az y′(x) = y(x)2 −(x + 1)y(x) + 1 diﬀerenciálegyenlet szukcesszív approximá- ciójával kapott első két közelítő függvényt, ha a kezdeti feltétel y(0) = 1. 2. Számoljuk ki az y′(x) = y(x) diﬀerenciálegyenlet szukcesszív approximációjával kapott első négy közelítő függvényt, ha a kezdeti feltétel y(0) = 1. 3. Oldjuk meg az y′ = e−y sin2 x diﬀerenciálegyenletet y(0) = 0 kezdeti feltétel mellett. 4. Egy test zuhan függőlegesen a gravitáció és a sebesség négyzetével arányos közegellenállás hatására. A mozgást az y : R →R magasság-idő-függvény írja le, ami eleget tesz a y′′(t) = −g + αy′(t)2 diﬀerenciálegyenletnek. A t = 0 pillanatban a test áll és y(0) = h magasan tartózkodik. Hogyan mozog ezután? 5. Oldjuk meg az xy′ = y + √x2 + y2 diﬀerenciálegyenletet y(3) = 4 kezdeti feltétel mellett. További gyakorló feladatok 6. Számoljuk ki az y′(x) = y(x) x diﬀerenciálegyenlet szukcesszív approximációjával kapott első három közelítő függvényt, ha a kezdeti feltétel y(1) = 1. 7. Szukcesszív approximáció segítségével határozzuk meg az y′(x) = x+y(x) diﬀerenciálegyen- let y(0) = 1 kezdeti feltételhez tartozó megoldását. 8. Határozzuk meg a (2x + 1)y′ −3y = 0 diﬀerenciálegyenlet általános megoldását. 9. Határozzuk meg az (1 + x2)y′ + (1 + y2) = 0 diﬀerenciálegyenlet általános megoldását. 10. Oldjuk meg az y′′ = −2xy′2 diﬀerenciálegyenletet y(0) = 0, y′(0) = 1 kezdeti feltétel mellett. 11. Oldjuk meg a 2xyy′ = y2 −x2 diﬀerenciálegyenletet y(1) = 1 kezdeti feltétellel. 12. Oldjuk meg az xy′ = y −x cos2 y x diﬀerenciálegyenletet.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 7. feladatsor: Szukcesszív approximáció, néhány egyenlettípus 1. Számoljuk ki az y[′](x) = y(x)[2] _−_ (x + 1)y(x) + 1 differenciálegyenlet szukcesszív approximációjával kapott első két közelítő függvényt, ha a kezdeti feltétel y(0) = 1. 2. Számoljuk ki az y[′](x) = y(x) differenciálegyenlet szukcesszív approximációjával kapott első négy közelítő függvényt, ha a kezdeti feltétel y(0) = 1. 3. Oldjuk meg az y[′] = e[−][y] sin[2] _x differenciálegyenletet y(0) = 0 kezdeti feltétel mellett._ 4. Egy test zuhan függőlegesen a gravitáció és a sebesség négyzetével arányos közegellenállás hatására. A mozgást az y : _→_ magasság-idő-függvény írja le, ami eleget tesz a R R _y[′′](t) = −g + αy[′](t)[2]_ differenciálegyenletnek. A t = 0 pillanatban a test áll és y(0) = h magasan tartózkodik. Hogyan mozog ezután? 5. Oldjuk meg az xy[′] = y + _[√]x[2]_ + y[2] differenciálegyenletet y(3) = 4 kezdeti feltétel mellett. ## További gyakorló feladatok 6. Számoljuk ki az y[′](x) = _[y][(][x][)]_ differenciálegyenlet szukcesszív approximációjával kapott első _x_ három közelítő függvényt, ha a kezdeti feltétel y(1) = 1. 7. Szukcesszív approximáció segítségével határozzuk meg az y[′](x) = x+y(x) differenciálegyenlet y(0) = 1 kezdeti feltételhez tartozó megoldását. 8. Határozzuk meg a (2x + 1)y[′] _−_ 3y = 0 differenciálegyenlet általános megoldását. 9. Határozzuk meg az (1 + x[2])y[′] + (1 + y[2]) = 0 differenciálegyenlet általános megoldását. 10. Oldjuk meg az y[′′] = −2xy[′][2] differenciálegyenletet y(0) = 0, y[′](0) = 1 kezdeti feltétel mellett. 11. Oldjuk meg a 2xyy[′] = y[2] _−_ _x[2]_ differenciálegyenletet y(1) = 1 kezdeti feltétellel. 12. Oldjuk meg az xy[′] = y − _x cos[2][ y]_ _x_ [differenciálegyenletet.] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">7. feladatsor: Szukcesszív approximáció, néhány</span></b></p> <p style="top:106.3pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">egyenlettípus</span></b></p> <p style="top:147.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Számoljuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet szukcesszív approximá-</span></p> <p style="top:161.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciójával kapott első két közelítő függvényt, ha a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:177.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Számoljuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet szukcesszív approximációjával kapott első</span></p> <p style="top:192.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">négy közelítő függvényt, ha a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:208.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:225.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Egy test zuhan függőlegesen a gravitáció és a sebesség négyzetével arányos közegellenállás</span></p> <p style="top:239.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hatására. A mozgást az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> magasság-idő-függvény írja le, ami eleget tesz a</span></p> <p style="top:266.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:292.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletnek. A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pillanatban a test áll és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> magasan tartózkodik.</span></p> <p style="top:306.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hogyan mozog ezután?</span></p> <p style="top:323.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(3) = 4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:356.1pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:380.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Számoljuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:387.5pt;left:209.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:380.2pt;left:224.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet szukcesszív approximációjával kapott első</span></p> <p style="top:394.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">három közelítő függvényt, ha a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:411.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Szukcesszív approximáció segítségével határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyen-</span></p> <p style="top:425.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">let</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételhez tartozó megoldását.</span></p> <p style="top:441.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:458.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Határozzuk meg az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:474.8pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:489.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:505.7pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Oldjuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétellel.</span></p> <p style="top:522.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup></p> <p style="top:529.5pt;left:240.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet.</span></sup></p> </div>	page_269.png	Matematika A3 gyakorlat Energetika és Mechatzonika TsSc szakok, 2016/17 ősz 7. feladatsor: Szukcesszív approximáció, néhány egyenlettípus 1. Számoljuk ki az y(z) — y(2?— (z-e1yyíz) 2-1 diflerene ciójával kapott első két közelítő függvényt, ha a kezdeti fel 1900)—1. közelítő függvényt, ha a kezdeti feltétel y(0) 3. Oldjuk meg az Y — e-! sit 2 diferenciálegyenletet y(0) — 0 kezdeti feltétel mellett. szív approxirm uhan v - —9 a dífferenciálegyenletnek. A t Hogyan mozog ezután? 5. Oldjuk meg az zy/ — y 4 VT FY dífferenciól 0 pillanatban a test áll és y(0) — A magasan tartózkodik. A kezdeti feltétel mellett. "További gyakorló feladatok 6. Számoljuk ki az v(z) három közelítő függvényt. ha a kezdeti felté 190)—1. let y(0) — 1 kezdeti feltételhi 8. Határozzuk meg a (2r 4. 1jy — 3y — 0 dilferenciálegyenlet általános megoldását 9. Határozzuk meg az (14-27)y 4 (1-4 1) — 0 differenciálegyenlet általános megoldását 2x dillerenciálegyenletet y(0) — 0. y(0) — 1 kezdeti feltétel mellett 11. Oldjuk meg a 2ryy — 4 — 2? differenciálegyenletet y(1) — 1 kezdeti feltétellel.
10. Oldjuk meg az y′′ = −2xy′2 diﬀerenciálegyenletet y(0) = 0, y′(0) = 1 kezdeti feltétel mellett. Megoldás. Vezessük be a v = y′ jelölést, erre a függvényre nézve az egyenlet elsőrendű: v′ = −2xv2, ami szétválasztható. A kezdeti feltétel v(0) = y′(0) = 1, tehát v′ v v2 = −2x Z x Z x v′(ξ) v(ξ)2 dξ = Z 0 (−2ξ) dξ −1 v(x) + 1 v(0) = −x2 v(x) = v(0) 1 + v(0)x2 = 1 1 + x2. Integrálással kapjuk a megoldást: Z x 0 v(ξ) dξ = y(x) = y(0) + Z x 1 1 + ξ2 dξ = arctan x. 11. Oldjuk meg a 2xyy′ = y2 −x2 diﬀerenciálegyenletet y(1) = 1 kezdeti feltétellel. Megoldás. A kezdeti feltétel környezetében 2xy ̸= 0, így eloszthatjuk vele az egyenletet: y′ = 1 2 2 y x −1 x y , a jobb oldal csak y/x-től függ, így y = xu helyettesítést alkalmazhatunk. Ekkor y′ = u+xu′, tehát u′ = y′ −u y 2x2 −1 2y 1 2y −u x u x = 1 x u u 2 −1 2u 1 2u −u = −1 2x = −1 2 u + 1 2uu′ 2uu′ 1 + u2 = −1 x 1 x. Mindkét oldalt integrálva x > 0 esetén ln x0 x = − ln x0 x Z x x0 Z u(x) u(x0) 2u 1 + u2du = u(x) u(x0) = ln 1 + u(x)2 1 + u(x0)2 i ln(1 + u2) iu(x) 1 ξ dξ = 1 + u(x0)2 tehát a megoldás y(x) = xu(x) = u u tx0 1 + y2 0 x2 0 x −x2 = √ 2x −x2. 12. Oldjuk meg az xy′ = y −x cos2 y x diﬀerenciálegyenletet. Megoldás. x ̸= 0 esetén az egyenlet 12. Oldjuk meg az xy′ = y −x cos2 y x y′ = y x y x −cos2 y x y x, ahol a jobb oldal csak y/x függvénye. y = xu helyettesítéssel y′ = u + xu′, tehát u′ = y′ −u x = y x2 y x2 −1 x 1 x cos2 y x y x −u x u x = −1 x 1 x cos2 u u′ cos2 u = −1 x u′ tan u(x) = −ln x + C y(x) = x arctan (C −ln x) .	10. Oldjuk meg az y[′′] = −2xy[′][2] differenciálegyenletet y(0) = 0, y[′](0) = 1 kezdeti feltétel mellett. _Megoldás. Vezessük be a v = y[′]_ jelölést, erre a függvényre nézve az egyenlet elsőrendű: _v[′]_ = −2xv[2], ami szétválasztható. A kezdeti feltétel v(0) = y[′](0) = 1, tehát _v[′]_ _v[2][ =][ −][2][x]_ � _x_ _v[′](ξ)_ � _x_ 0 _v(ξ)[2][ d][ξ][ =]_ 0 [(][−][2][ξ][) d][ξ] 1 _−_ [1] _v(x) [+]_ _v(0) [=][ −][x][2]_ _v(0)_ 1 _v(x) =_ 1 + v(0)x[2][ =] 1 + x[2] _[.]_ Integrálással kapjuk a megoldást: � _x_ � _x_ 1 _y(x) = y(0) +_ 0 _[v][(][ξ][) d][ξ][ =]_ 0 1 + ξ[2][ d][ξ][ = arctan][ x.] 11. Oldjuk meg a 2xyy[′] = y[2] _−_ _x[2]_ differenciálegyenletet y(1) = 1 kezdeti feltétellel. _Megoldás. A kezdeti feltétel környezetében 2xy ̸= 0, így eloszthatjuk vele az egyenletet:_ _y_ _x_ _y[′]_ = [1]2 _x_ _[−]_ [1]2 _y [,]_ a jobb oldal csak y/x-től függ, így y = xu helyettesítést alkalmazhatunk. Ekkor y[′] = u+xu[′], tehát _u[′]_ = _[y][′][ −]_ _[u]_ = _y_ � _u_ � = _[−][1]_ �u + [1] � _x_ 2x[2][ −] 2[1]y _[−]_ _[u]x_ [= 1]x 2 _[−]_ 2[1]u _[−]_ _[u]_ 2x _u_ 2uu[′] 1 + u[2][ =][ −]x[1] _[.]_ Mindkét oldalt integrálva x > 0 esetén ln _[x][0]_ � _x_ 1 � _u(x)_ 2u �ln(1 + u[2])�u(x) _x_ [=][ −] _x0_ _ξ [dξ][ =]_ _u(x0)_ 1 + u[2] _[du][ =]_ _u(x0)_ [= ln 1 +]1 + u[ u]([(]x[x]0[)])[2][2] tehát a megoldás _y(x) = xu(x) =_ � � ��x0 � 1 + _[y]0[2]_ _x[2]0_ 2x − _x[2]._ � _√_ _x −_ _x[2]_ = 12. Oldjuk meg az xy[′] = y − _x cos[2][ y]_ _x_ [differenciálegyenletet.] _Megoldás. x ̸= 0 esetén az egyenlet_ _y[′]_ = _[y]_ _x_ _[−]_ [cos][2][ y]x _[,]_ ahol a jobb oldal csak y/x függvénye. y = xu helyettesítéssel y[′] = u + xu[′], tehát _u[′]_ = _[y][′][ −]_ _[u]_ = _[y]_ _x_ _x[2][ −]_ _x[1]_ [cos][2][ y]x _[−]_ _[u]x_ [=][ −]x[1] [cos][2][ u] _u[′]_ cos[2] _u_ [=][ −]x[1] tan u(x) = − ln x + C _y(x) = x arctan (C −_ ln x) . -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:91.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Vezessük be a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">jelölést, erre a függvényre nézve az egyenlet elsőrendű:</span></p> <p style="top:105.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xv</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami szétválasztható. A kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:125.8pt;left:166.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:142.0pt;left:165.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:150.6pt;left:118.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:174.9pt;left:124.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:155.0pt;left:137.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:171.3pt;left:136.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:150.6pt;left:193.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:174.9pt;left:199.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup></p> <p style="top:193.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:202.1pt;left:116.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:185.8pt;left:163.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:202.1pt;left:155.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:226.1pt;left:156.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:218.0pt;left:210.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span></p> <p style="top:234.3pt;left:194.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:218.0pt;left:278.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:234.3pt;left:265.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:254.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Integrálással kapjuk a megoldást:</span></p> <p style="top:280.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) +</span></p> <p style="top:268.0pt;left:179.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:292.3pt;left:185.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:268.0pt;left:247.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:292.3pt;left:253.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:272.4pt;left:278.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:288.7pt;left:266.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = arctan</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></sup></p> <p style="top:309.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Oldjuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xyy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétellel.</span></p> <p style="top:326.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A kezdeti feltétel környezetében</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, így eloszthatjuk vele az egyenletet:</span></p> <p style="top:353.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:361.2pt;left:132.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:344.9pt;left:140.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:361.2pt;left:140.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:361.2pt;left:164.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:344.9pt;left:172.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:361.2pt;left:172.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:381.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a jobb oldal csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y/x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től függ, így</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítést alkalmazhatunk. Ekkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xu</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:395.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span></p> <p style="top:420.8pt;left:131.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup></p> <p style="top:429.0pt;left:169.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:420.8pt;left:192.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:412.7pt;left:211.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:429.0pt;left:205.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:429.0pt;left:240.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup></p> <p style="top:429.0pt;left:269.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></sup></p> <p style="top:429.0pt;left:293.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:407.8pt;left:303.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:429.0pt;left:312.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:429.0pt;left:335.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup></p> <p style="top:407.8pt;left:370.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:420.8pt;left:381.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:429.0pt;left:396.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:407.8pt;left:413.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:420.8pt;left:420.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:429.0pt;left:443.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:407.8pt;left:450.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:444.0pt;left:112.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">uu</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:460.3pt;left:107.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:460.3pt;left:166.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:478.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mindkét oldalt integrálva</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén</span></p> <p style="top:506.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup></p> <p style="top:514.6pt;left:121.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:493.9pt;left:159.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:518.2pt;left:164.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span></p> <p style="top:498.3pt;left:177.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:514.6pt;left:177.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:493.9pt;left:212.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:518.2pt;left:217.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:498.3pt;left:251.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:514.6pt;left:241.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">du</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:496.4pt;left:303.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">h</span></p> <p style="top:506.4pt;left:308.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:496.4pt;left:358.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">i</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:516.1pt;left:363.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= ln 1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:514.6pt;left:412.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:534.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a megoldás</span></p> <p style="top:567.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:547.5pt;left:188.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">v</span></p> <p style="top:553.1pt;left:188.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">u</span></p> <p style="top:559.1pt;left:188.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">u</span></p> <p style="top:565.1pt;left:188.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">t</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:551.6pt;left:212.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span></p> <p style="top:567.5pt;left:220.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:565.2pt;left:248.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:575.7pt;left:242.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:581.5pt;left:248.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:551.6pt;left:254.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:567.5pt;left:264.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:557.1pt;left:313.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:567.5pt;left:323.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:599.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup></p> <p style="top:606.7pt;left:240.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet.</span></sup></p> <p style="top:616.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén az egyenlet</span></p> <p style="top:641.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:649.2pt;left:132.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup></p> <p style="top:649.2pt;left:178.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:666.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol a jobb oldal csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y/x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvénye.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítéssel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:695.0pt;left:138.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup></p> <p style="top:703.2pt;left:176.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:695.0pt;left:199.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:703.2pt;left:213.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:703.2pt;left:241.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup></p> <p style="top:703.2pt;left:274.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup></p> <p style="top:703.2pt;left:298.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:703.2pt;left:332.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup></p> <p style="top:716.0pt;left:127.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:732.3pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:732.3pt;left:174.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:746.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></p> <p style="top:763.8pt;left:125.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> arctan (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> .</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_270.png	10. Oldjuk meg az 4" — —2247 differenciálegyenletet y(0) — 0. y(0) — 1 kezdeti feltétel mellett feltétel (0) — (0) — 1, tehát hzgyé s [/Czds 2 , 1 mo tecoi e) 1 o- Integrálással kapjuk a megoldást. 910094 [£ ( eh artan 1 kezdeti feltétellel. előszthatjuk vele az egyet 11. Oldjuk meg a 2ryy/ — 4 — 2? dífferenciálegyenletet y( .Megoldás. A kezdi tehát u y 1 EEET] 1 Mindkét oldalt integrálva 2 —. 0 csetén m. 1 l9 24 age — , 1-4-u(2)? hE főté [ődcAuthh] O E tehát a megoldás wía) — zulz) 12. Oldjuk meg az 2 — y — zeos? ! diferenciálegyenletet .Megoldás, 2 74 0 eset peYze o 1 ogy 92 1l w o tanu(r) ah wle) — raretan (C — In2)
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 7. feladatsor: Szukcesszív approximáció, néhány egyenlettípus (megoldás) 1. Számoljuk ki az y′(x) = y(x)2 −(x + 1)y(x) + 1 diﬀerenciálegyenlet szukcesszív approximá- ciójával kapott első két közelítő függvényt, ha a kezdeti feltétel y(0) = 1. Megoldás. A 0. függvény konstans ϕ0(x) = y(0) = 1, a függvénysorozat többi tagját rekurzióval deﬁniáljuk: ϕk+1(x) = 1 + Z x 0 (ϕk(ξ)2 −(ξ + 1)ϕk(ξ) + 1) dξ, ennek alapján ϕ0(x) = 1 ϕ1(x) = 1 + x −x2 2 ϕ2(x) = 1 + x −x3 6 x3 6 −x4 8 x4 8 + x5 20 x 20. 2. Számoljuk ki az y′(x) = y(x) diﬀerenciálegyenlet szukcesszív approximációjával kapott első négy közelítő függvényt, ha a kezdeti feltétel y(0) = 1. Megoldás. A ϕ0(x) = y(0) = 1 függvényből kiindulva ϕk+1(x) = 1 + Z x Z 0 ϕk(ξ) dξ, ennek alapján a függvénysorozat első négy tagja ϕ0(x) = 1 ϕ1(x) = 1 + x ϕ2(x) = 1 + x + x2 2 ϕ3(x) = 1 + x + x2 2 x2 2 + x3 6 ϕ4(x) = 1 + x + x2 2 x2 2 + x3 6 x3 6 + x4 24 x 24. Ebből egyébként rájöhetünk az általános tag alakjára: ϕk(x) = n=0 xn n! , ezt teljes indukcióval lehet bizonyítani. A kapott függvénysorozat minden korlátos intervallumon egyenletesen konvergál az y(x) = ex függvényhez, ez megoldja a Cauchy-feladatot. 3. Oldjuk meg az y′ = e−y sin2 x diﬀerenciálegyenletet y(0) = 0 kezdeti feltétel mellett. Megoldás. Az egyenlet szétválasztható, ráadásul e−y sehol sem 0, tehát mindkét oldalt eloszthatjuk vele: eyy′ = sin2 x. Ezután integráljuk mindkét oldalt x0 = 0-tól x-ig: ezt teljes indukcióval lehet bizonyítani. A kapott függvénysorozat minden korlátos intervallumon egyenletesen konvergál az y(x) = ex függvényhez, ez megoldja a Cauchy-feladatot. 3. Oldjuk meg az y′ = e−y sin2 x diﬀerenciálegyenletet y(0) = 0 kezdeti feltétel mellett. Z x 0 ey(ξ)y′(ξ) dξ = Z x Z x 0 sin2 ξ dξ = Z x Z x 1 −cos 2ξ dξ "ξ 2 −sin 2ξ 4 #ξ=x ξ=0 ey(x) −ey(0) = "ξ ey(x) −1 = x 2 x 2 −1 4 1 4 sin 2x.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 7. feladatsor: Szukcesszív approximáció, néhány egyenlettípus (megoldás) 1. Számoljuk ki az y[′](x) = y(x)[2] _−_ (x + 1)y(x) + 1 differenciálegyenlet szukcesszív approximációjával kapott első két közelítő függvényt, ha a kezdeti feltétel y(0) = 1. _Megoldás. A 0._ függvény konstans ϕ0(x) = y(0) = 1, a függvénysorozat többi tagját rekurzióval definiáljuk: � _x_ _ϕk+1(x) = 1 +_ 0 [(][ϕ][k][(][ξ][)][2][ −] [(][ξ][ + 1)][ϕ][k][(][ξ][) + 1) d][ξ,] ennek alapján _ϕ0(x) = 1_ _ϕ1(x) = 1 + x −_ _[x][2]_ 2 _ϕ2(x) = 1 + x −_ _[x][3]_ 6 _[−]_ _[x]8 [4]_ [+][ x]20[5] _[.]_ 2. Számoljuk ki az y[′](x) = y(x) differenciálegyenlet szukcesszív approximációjával kapott első négy közelítő függvényt, ha a kezdeti feltétel y(0) = 1. _Megoldás. A ϕ0(x) = y(0) = 1 függvényből kiindulva_ � _x_ _ϕk+1(x) = 1 +_ 0 _[ϕ][k][(][ξ][) d][ξ,]_ ennek alapján a függvénysorozat első négy tagja _ϕ0(x) = 1_ _ϕ1(x) = 1 + x_ _ϕ2(x) = 1 + x +_ _[x][2]_ 2 _ϕ3(x) = 1 + x +_ _[x][2]_ 2 [+][ x]6[3] _ϕ4(x) = 1 + x +_ _[x][2]_ 2 [+][ x]6 [3] [+][ x]24[4] _[.]_ Ebből egyébként rájöhetünk az általános tag alakjára: _ϕk(x) =_ _k_ � _n=0_ _x[n]_ _n!_ _[,]_ ezt teljes indukcióval lehet bizonyítani. A kapott függvénysorozat minden korlátos intervallumon egyenletesen konvergál az y(x) = e[x] függvényhez, ez megoldja a Cauchy-feladatot. 3. Oldjuk meg az y[′] = e[−][y] sin[2] _x differenciálegyenletet y(0) = 0 kezdeti feltétel mellett._ _Megoldás. Az egyenlet szétválasztható, ráadásul e[−][y]_ sehol sem 0, tehát mindkét oldalt eloszthatjuk vele: e[y]y[′] = sin[2] _x. Ezután integráljuk mindkét oldalt x0 = 0-tól x-ig:_ � _x_ � _x_ � _x_ 1 − cos 2ξ dξ 0 _[e][y][(][ξ][)][y][′][(][ξ][) d][ξ][ =]_ 0 [sin][2][ ξ][ d][ξ][ =] 0 2 � _ξ_ �ξ=x _e[y][(][x][)]_ _−_ _e[y][(0)]_ = 2 _[−]_ [sin 2]4 _[ξ]_ _ξ=0_ _e[y][(][x][)]_ _−_ 1 = _[x]_ 2 _[−]_ 4 [1] [sin 2][x.] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">7. feladatsor: Szukcesszív approximáció, néhány</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">egyenlettípus (megoldás)</span></b></p> <p style="top:142.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Számoljuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet szukcesszív approximá-</span></p> <p style="top:156.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciójával kapott első két közelítő függvényt, ha a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:173.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A 0.</span></p> <p style="top:173.2pt;left:167.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvény konstans</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a függvénysorozat többi tagját</span></p> <p style="top:187.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rekurzióval deﬁniáljuk:</span></p> <p style="top:211.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span></p> <p style="top:198.5pt;left:181.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:222.7pt;left:187.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 1) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ,</span></i></sup></p> <p style="top:234.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek alapján</span></p> <p style="top:253.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></p> <p style="top:279.5pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:287.7pt;left:197.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:309.2pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:317.4pt;left:197.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:317.4pt;left:226.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup></p> <p style="top:317.4pt;left:251.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">20</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:333.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Számoljuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet szukcesszív approximációjával kapott első</span></p> <p style="top:347.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">négy közelítő függvényt, ha a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:364.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvényből kiindulva</span></p> <p style="top:388.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span></p> <p style="top:375.9pt;left:181.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:400.1pt;left:187.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ,</span></i></sup></p> <p style="top:411.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek alapján a függvénysorozat első négy tagja</span></p> <p style="top:431.0pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></p> <p style="top:448.4pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:474.3pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:482.5pt;left:197.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:504.0pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:512.2pt;left:197.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:512.2pt;left:225.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:533.8pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:542.0pt;left:197.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:542.0pt;left:225.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:542.0pt;left:251.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">24</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:556.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből egyébként rájöhetünk az általános tag alakjára:</span></p> <p style="top:585.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:575.7pt;left:156.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:577.1pt;left:151.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:600.4pt;left:150.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:577.4pt;left:169.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup></p> <p style="top:593.6pt;left:170.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">!</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:613.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezt teljes indukcióval lehet bizonyítani. A kapott függvénysorozat minden korlátos interval-</span></p> <p style="top:627.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lumon egyenletesen konvergál az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvényhez, ez megoldja a Cauchy-feladatot.</span></p> <p style="top:644.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:661.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet szétválasztható, ráadásul</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sehol sem</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát mindkét oldalt</span></p> <p style="top:675.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">eloszthatjuk vele:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ezután integráljuk mindkét oldalt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-tól</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ig:</span></p> <p style="top:688.5pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:712.7pt;left:112.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:688.5pt;left:198.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:712.7pt;left:203.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:688.5pt;left:272.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:712.7pt;left:277.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:692.9pt;left:290.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></p> <p style="top:709.2pt;left:312.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:701.0pt;left:343.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></p> <p style="top:733.8pt;left:125.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(0)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:717.9pt;left:198.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></p> <p style="top:742.0pt;left:205.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup></p> <p style="top:742.0pt;left:239.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:717.9pt;left:257.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:749.5pt;left:263.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:767.4pt;left:140.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:775.6pt;left:199.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:775.6pt;left:223.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></sup></p> </div>	page_271.png	Matematika A3 gyakorlat Energetika és Mechatzonika TsSc szakok, 2016/17 ősz 7. feladatsor: Szukcesszív approximáció, néhány egyenlettípus (megoldás) 1. Számoljuk ki az y(z) — y(z? — (2-1yíz) 2-1 dillerenciáleg ciójával kapott első két közelítő füi enlet szukcesszív approximá- tvényt, ha a kezdeti feltétel y(0) — 1. Megoldás. A 0. függyé rekurzióval definiáljuk: konstans solr) — y(0) — 1. a fűggvénysorozat többi tagját erala) 14 [ot ]t 4 1Ah 41946. ennek alapján. ede) -1 eldelér § edralár -t közelítő függvényt, ha a kezdeti feltétel y(0) . Megoldás. A slz) — v(0) függvényből kiindulva eealt) 14 [/eloas lo -1 e -ltz eloltri elelért ÉrE 1ér tt a0-E- korlátos interval- ezt teljes indukcióval lehet bizonyítani. A kapott függvénysorozat minde Iumon egyenletesen könvergál az y(z) — e" függvényhez, ez megoldja a Cauchy-feladatot. 3. Oldjuk meg az Y letet y(0) — 0 kezdeti feltétel mellett. Megoldás. Az egyenlet szétválasztható, ráadásul €-? sehol sem 0, tehát eloszthatjuk vele: elyi — sín? 2. Ezután integráljuk mindkét oldalt 24 — 0-tól 2-i [o e em — sín? s differenciálegyet c0s2£ , 2 e0-1
Ebből y(x) kifejezhető: y(x) = ln 1 + x 2 x 2 −1 4 1 4 sin 2x 4. Egy test zuhan függőlegesen a gravitáció és a sebesség négyzetével arányos közegellenállás hatására. A mozgást az y : R →R magasság-idő-függvény írja le, ami eleget tesz a y′′(t) = −g + αy′(t)2 diﬀerenciálegyenletnek. A t = 0 pillanatban a test áll és y(0) = h magasan tartózkodik. Hogyan mozog ezután? Megoldás. v = y′ helyettesítéssel v′ = −g + αv2 szétválasztható, a megoldása artanh qα α g v(t) Z t 0 dτ = Z t Z t Z v(t) t = v′(τ) αv(τ)2 −gdτ = 1 αν2 −gdν = − g √αg amiből v(t) meghatározható: g r g v(t) = − g α tanh (√αgt) ezt integrálva y(t) = h + = h − Z t r g Z t Z 0 v(τ)dτ = h − Z t 0 tanh(√αgτ)dτ α "ln cosh(√αgt) √αg 0 r g #τ=t τ=0 = h −1 α ln cosh(t√gα) = h −1 α Érdemes a kapott függvényt összehasonlítani a közegellenállás nélküli zuhanást leíró jól ismert h −g 2t2 függvénnyel. Ehhez fejtsük sorba a megoldást t = 0 körül (vagy α = 0 körül α szerint, ez ugyanazt adja): Érdemes a kapott függvényt összehasonlítani a közegellenállás nélküli zuhanást leíró jól ismert h −g 2t2 függvénnyel. Ehhez fejtsük sorba a megoldást t = 0 körül (vagy α = 0 körül h −g 2 g 2t2 + g2α 12 g2α 45 12 t4 −g3α2 g3α2 45 t6 + . . . . 2 12 45 5. Oldjuk meg az xy′ = y + √x2 + y2 diﬀerenciálegyenletet y(3) = 4 kezdeti feltétel mellett. Megoldás. Osszuk el mindkét oldalt x-szel (nem 0 az x0 = 3 pont egy környezetében): 2 y′ = y x + 1 x y′ = y x q x2 + y2 = y x + q x2 + y2 = y x 1 + y tehát a jobb oldal csak y/x-től függ. u = y x helyettesítéssel és xu = y, u + xu′ = y′ felhasználásával x > 0 esetén az tehát a jobb oldal csak y/x-től függ. u = y x helyettesítéssel és xu = y, u + xu′ = y′ f lh álá á l 0 té u + xu′ = u + √ 1 + u2 egyenletet kapjuk, amiből u′ u′ √ x 1 + u2 = 1 Z x Z x u′(ξ) q 1 + u(ξ)2 dξ = u′(ξ) 1 ξ dξ arsinh u(x) −arsinh 4 3 = ln x 3 arsinh u(x) −arsinh 4 3 és így a megoldás y(x) = xu(x) = x sinh arsinh 4 3 + ln x 3	Ebből y(x) kifejezhető: � � _y(x) = ln_ 1 + _[x]2_ _[−]_ [1]4 [sin 2][x] _._ 4. Egy test zuhan függőlegesen a gravitáció és a sebesség négyzetével arányos közegellenállás hatására. A mozgást az y : _→_ magasság-idő-függvény írja le, ami eleget tesz a R R _y[′′](t) = −g + αy[′](t)[2]_ differenciálegyenletnek. A t = 0 pillanatban a test áll és y(0) = h magasan tartózkodik. Hogyan mozog ezután? _Megoldás. v = y[′]_ helyettesítéssel v[′] = −g + αv[2] szétválasztható, a megoldása �� _α_ � � _t_ � _t_ _v[′](τ_ ) � _v(t)_ 1 artanh _g_ _[v][(][t][)]_ _t =_ _√_ _,_ 0 _[dτ][ =]_ 0 _αv(τ_ )[2] _−_ _g_ _[dτ][ =]_ 0 _αν[2]_ _−_ _g_ _[dν][ =][ −]_ _αg_ amiből v(t) meghatározható: � _g_ _v(t) = −_ _α_ [tanh (][√][αgt][)] ezt integrálva � _t_ � _g_ � _t_ _y(t) = h +_ 0 _[v][(][τ]_ [)][dτ][ =][ h][ −] _α_ 0 [tanh(][√][αgτ] [)][dτ] � _g_ � ln cosh(√αgt) �τ =t = h − _α_ _√αg_ = h − _α[1]_ [ln cosh(][t][√][gα][)] _τ_ =0 Érdemes a kapott függvényt összehasonlítani a közegellenállás nélküli zuhanást leíró jól ismert h − _[g]_ 2 _[t][2][ függvénnyel. Ehhez fejtsük sorba a megoldást][ t][ = 0 körül (vagy][ α][ = 0 körül]_ _α szerint, ez ugyanazt adja):_ _h −_ _[g]_ 2 _[t][2][ +][ g]12[2][α]_ _[t][4][ −]_ _[g]45[3][α][2]_ _[t][6][ +][ . . . .]_ 5. Oldjuk meg az xy[′] = y + _[√]x[2]_ + y[2] differenciálegyenletet y(3) = 4 kezdeti feltétel mellett. _Megoldás. Osszuk el mindkét oldalt x-szel (nem 0 az x0 = 3 pont egy környezetében):_ _y[′]_ = _x[y]_ [+ 1]x � � _y_ 1 + _x_ �2 _,_ � _x[2]_ + y[2] = _[y]_ _x_ [+] tehát a jobb oldal csak y/x-től függ. _u =_ _y_ _x_ [helyettesítéssel és][ xu][ =][ y][,][ u][ +][ xu][′][ =][ y][′] felhasználásával x > 0 esetén az _√_ _u + xu[′]_ = u + 1 + u[2] egyenletet kapjuk, amiből _u[′]_ _√_ 1 + u[2][ = 1]x � _x_ _u[′](ξ)_ � _x_ 1 3 �1 + u(ξ)[2][ d][ξ][ =] 3 _ξ_ [d][ξ] arsinh u(x) − arsinh [4] 3 [= ln][ x]3 és így a megoldás � _y(x) = xu(x) = x sinh_ arsinh [4] 3 [+ ln][ x]3 � _._ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kifejezhető:</span></p> <p style="top:85.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = ln</span></p> <p style="top:72.5pt;left:155.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:85.4pt;left:163.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:93.6pt;left:185.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:93.6pt;left:208.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:72.5pt;left:246.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:85.4pt;left:255.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:110.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Egy test zuhan függőlegesen a gravitáció és a sebesség négyzetével arányos közegellenállás</span></p> <p style="top:124.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hatására. A mozgást az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> :</span><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000"> R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> magasság-idő-függvény írja le, ami eleget tesz a</span></p> <p style="top:146.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:167.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletnek. A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pillanatban a test áll és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> magasan tartózkodik.</span></p> <p style="top:181.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hogyan mozog ezután?</span></p> <p style="top:198.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyettesítéssel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> αv</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szétválasztható, a megoldása</span></p> <p style="top:237.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:224.7pt;left:126.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> t</span></i></p> <p style="top:248.9pt;left:132.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dτ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:224.7pt;left:170.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> t</span></i></p> <p style="top:248.9pt;left:175.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:229.1pt;left:201.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">τ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:245.4pt;left:186.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αv</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">τ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dτ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:224.7pt;left:270.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:248.9pt;left:276.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:229.1pt;left:315.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:245.4pt;left:298.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αν</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dν</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:223.8pt;left:378.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">artanh</span></p> <p style="top:210.8pt;left:413.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></p> <p style="top:231.1pt;left:433.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">g</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:210.8pt;left:458.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:238.0pt;left:410.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αg</span></i></p> <p style="top:237.1pt;left:467.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:266.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> meghatározható:</span></p> <p style="top:292.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:279.2pt;left:150.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">r</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> g</span></i></p> <p style="top:301.0pt;left:162.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tanh (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αgt</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:317.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezt integrálva</span></p> <p style="top:343.1pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:330.6pt;left:164.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> t</span></i></p> <p style="top:354.9pt;left:170.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">τ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dτ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:329.6pt;left:251.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">r</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> g</span></i></p> <p style="top:351.3pt;left:262.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:330.6pt;left:273.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> t</span></i></p> <p style="top:354.9pt;left:278.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tanh(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αgτ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dτ</span></i></sup></p> <p style="top:375.3pt;left:131.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:361.8pt;left:165.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">r</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> g</span></i></p> <p style="top:383.5pt;left:176.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:359.4pt;left:186.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln cosh(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αgt</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:376.1pt;left:217.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αg</span></i></p> <p style="top:359.4pt;left:266.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">τ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></p> <p style="top:391.3pt;left:271.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">τ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:375.3pt;left:291.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:383.5pt;left:326.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln cosh(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">gα</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:406.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Érdemes a kapott függvényt összehasonlítani a közegellenállás nélküli zuhanást leíró jól</span></p> <p style="top:421.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ismert</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> h</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">g</span></i></sup></p> <p style="top:428.6pt;left:134.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvénnyel. Ehhez fejtsük sorba a megoldást</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> körül (vagy</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> körül</span></sup></p> <p style="top:435.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szerint, ez ugyanazt adja):</span></p> <p style="top:464.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">h</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i></sup></p> <p style="top:472.2pt;left:129.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> g</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup></p> <p style="top:472.2pt;left:164.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">g</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:472.2pt;left:210.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">45</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> . . . .</span></i></sup></p> <p style="top:490.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(3) = 4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:507.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Osszuk el mindkét oldalt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-szel (nem</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pont egy környezetében):</span></p> <p style="top:538.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:546.5pt;left:132.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span></sup></p> <p style="top:546.5pt;left:155.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:527.1pt;left:163.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:538.3pt;left:173.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:546.5pt;left:227.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:520.3pt;left:249.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">s</span></p> <p style="top:538.3pt;left:259.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></p> <p style="top:525.3pt;left:279.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:546.5pt;left:288.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:525.3pt;left:296.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:538.3pt;left:308.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:564.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a jobb oldal csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y/x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től függ.</span></p> <p style="top:564.9pt;left:282.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:562.9pt;left:311.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:572.2pt;left:311.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyettesítéssel és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:579.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felhasználásával</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén az</span></p> <p style="top:600.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:590.1pt;left:180.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:600.5pt;left:190.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:621.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletet kapjuk, amiből</span></p> <p style="top:640.5pt;left:191.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:648.0pt;left:175.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:657.8pt;left:185.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup></p> <p style="top:656.8pt;left:235.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:668.0pt;left:128.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:692.3pt;left:133.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:672.4pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:681.3pt;left:146.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:691.7pt;left:156.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:668.0pt;left:234.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:692.3pt;left:239.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:672.4pt;left:252.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:688.7pt;left:252.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup></p> <p style="top:716.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arsinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arsinh </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup></p> <p style="top:725.0pt;left:211.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:725.0pt;left:247.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:741.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és így a megoldás</span></p> <p style="top:766.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span></p> <p style="top:753.7pt;left:220.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:766.7pt;left:227.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arsinh </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup></p> <p style="top:774.9pt;left:262.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:774.9pt;left:297.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:753.7pt;left:304.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:766.7pt;left:314.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_272.png	Ebből yíz) kifejezhető: v - —9 a differenciálegyenletnek. A t — 0 pillanatban a test áll és y(0) — A magasan tartózkodik .Megoldás. 1 — 4 helyettesítéssel 1/ — —g 4 c7 szétválasztható, a megoldása eee [/ artanb ( /5e0) k JETOTET] a Wo vl anh(Yőg0) W0-A4 [/eode h x/É oc ográr 7 [dncosh(yag0917 1 2h [Z [EoA£uttvt] 2 nmeoshítyő) vr Z lncoshítygn) ismert A — 21 függvénnyel. Ehhez fejtsük sorba a megoldást t — 0 körül (vagy a — 0 körül 9a , Fan A hozít t s t 5. Oldjuk meg az 24 — y 4 V7F47 dílferenciálegy: v át a jobb oldal csak y/2-től függ. u — 2 helyettesítéssel és ru — y. u 4 w felhasználásával x -. 0 esetén az MYE etet kapjuk, amiből egyes 4 azsinhu(z) — arsính és így a megoldás
További gyakorló feladatok 6. Számoljuk ki az y′(x) = y(x) x diﬀerenciálegyenlet szukcesszív approximációjával kapott első három közelítő függvényt, ha a kezdeti feltétel y(1) = 1. Megoldás. A kezdeti függvény konstans ϕ0(x) = 1, a továbbiakat ϕk+1(x) = 1 + Z x ϕk(ξ) dξ módon deﬁniáljuk. Ebből ϕ0(x) = 1 ϕ1(x) = 1 + ln x ϕ2(x) = 1 + ln x + (ln x)2 2 ϕ3(x) = 1 + ln x + (ln x)2 2 + (ln x)3 6 adódik, ami alapján rájöhetünk, hogy ϕk(x) = n=0 (ln x)n n! , ennek limesze (x > 0 esetén) y(x) = e(ln x) = x, ami megoldja a kezdetiérték-problémát. 7. Szukcesszív approximáció segítségével határozzuk meg az y′(x) = x+y(x) diﬀerenciálegyen- let y(0) = 1 kezdeti feltételhez tartozó megoldását. Megoldás. A függvénysorozat 0. tagja ϕ0(x) = 1, a továbbiakat a ϕk+1(x) = 1 + Z x 0 (ξ + ϕk(ξ)) dξ rekurzió határozza meg. Az első néhány függvény ϕ0(x) = 1 ϕ1(x) = 1 + x + x2 2 ϕ2(x) = 1 + x + x2 + x3 6 ϕ3(x) = 1 + x + x2 + x3 3 x3 3 + x4 24 24 ϕ4(x) = 1 + x + x2 + x3 3 x3 3 + x4 12 x4 12 + x5 120 x 120, ebből sejthetjük, hogy xi ϕk(x) = −1 −x + 2 i=0 xi i! + xk+1 (k + 1 x (k + 1)!. Valóban: Z x 0 (ξ + ϕk(ξ))dξ = 1 + Z x 0 (−1 + 2 Z x ξi 1 + i=0 ξi i! + ξk+1 (k + 1 ξ (k + 1)!)dξ xi+1 (i + 1)! + xk+2 (k + 2 xi+1 = 1 −x + 2 = 1 −x + 2 (k + 2)! i=0 k+1 X i=1 xi xi i! + xk+2 (k + 2 x (k + 2)! = ϕk+1(x).	## További gyakorló feladatok 6. Számoljuk ki az y[′](x) = _[y][(][x][)]_ differenciálegyenlet szukcesszív approximációjával kapott első _x_ három közelítő függvényt, ha a kezdeti feltétel y(1) = 1. _Megoldás. A kezdeti függvény konstans ϕ0(x) = 1, a továbbiakat_ � _x_ _ϕk(ξ)_ _ϕk+1(x) = 1 +_ dξ 1 _ξ_ módon definiáljuk. Ebből _ϕ0(x) = 1_ _ϕ1(x) = 1 + ln x_ _ϕ2(x) = 1 + ln x + [(ln][ x][)][2]_ 2 _ϕ3(x) = 1 + ln x + [(ln][ x][)][2]_ + [(ln][ x][)][3] 2 6 adódik, ami alapján rájöhetünk, hogy _ϕk(x) =_ _k_ � _n=0_ (ln x)[n] _,_ _n!_ ennek limesze (x > 0 esetén) y(x) = e[(ln][ x][)] = x, ami megoldja a kezdetiérték-problémát. 7. Szukcesszív approximáció segítségével határozzuk meg az y[′](x) = x+y(x) differenciálegyenlet y(0) = 1 kezdeti feltételhez tartozó megoldását. _Megoldás. A függvénysorozat 0. tagja ϕ0(x) = 1, a továbbiakat a_ � _x_ _ϕk+1(x) = 1 +_ 0 [(][ξ][ +][ ϕ][k][(][ξ][)) d][ξ] rekurzió határozza meg. Az első néhány függvény _ϕ0(x) = 1_ _ϕ1(x) = 1 + x +_ _[x][2]_ 2 _ϕ2(x) = 1 + x + x[2]_ + _[x][3]_ 6 _ϕ3(x) = 1 + x + x[2]_ + _[x][3]_ 3 [+][ x]24[4] _ϕ4(x) = 1 + x + x[2]_ + _[x][3]_ 3 [+][ x]12 [4] [+][ x]120[5] _[,]_ ebből sejthetjük, hogy _ϕk(x) = −1 −_ _x + 2_ Valóban: _k_ � _i=0_ _x[i]_ _x[k][+1]_ _i! [+]_ (k + 1)![.] � _x_ � _x_ 1 + 0 [(][ξ][ +][ ϕ][k][(][ξ][))][dξ][ = 1 +] 0 [(][−][1 + 2] _k_ � _i=0_ _ξ[i]_ _ξ[k][+1]_ _i! [+]_ (k + 1)![)][dξ] = 1 − _x + 2_ = 1 − _x + 2_ _k_ � _i=0_ _k+1_ � _i=1_ _x[i][+1]_ _x[k][+2]_ (i + 1)! [+] (k + 2)! _x[i]_ _x[k][+2]_ _i! [+]_ (k + 2)! [=][ ϕ][k][+1][(][x][)][.] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:57.2pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:81.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Számoljuk ki az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:88.6pt;left:209.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:81.3pt;left:224.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet szukcesszív approximációjával kapott első</span></p> <p style="top:95.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">három közelítő függvényt, ha a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:113.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A kezdeti függvény konstans</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a továbbiakat</span></p> <p style="top:141.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span></p> <p style="top:128.9pt;left:181.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:153.2pt;left:187.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:133.3pt;left:200.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:149.6pt;left:211.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></p> <p style="top:141.4pt;left:231.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></p> <p style="top:168.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">módon deﬁniáljuk. Ebből</span></p> <p style="top:189.1pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></p> <p style="top:206.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 + ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:232.4pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 + ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:240.6pt;left:219.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:262.2pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 + ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:270.4pt;left:219.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:262.2pt;left:242.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:270.4pt;left:269.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:286.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik, ami alapján rájöhetünk, hogy</span></p> <p style="top:317.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:307.2pt;left:156.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:308.6pt;left:151.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:331.9pt;left:150.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:308.9pt;left:169.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup></p> <p style="top:325.2pt;left:181.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">!</span></p> <p style="top:317.0pt;left:204.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:348.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek limesze (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(ln</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami megoldja a kezdetiérték-problémát.</span></p> <p style="top:365.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Szukcesszív approximáció segítségével határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyen-</span></p> <p style="top:380.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">let</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételhez tartozó megoldását.</span></p> <p style="top:397.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A függvénysorozat 0. tagja</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a továbbiakat a</span></p> <p style="top:423.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span></p> <p style="top:410.7pt;left:181.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:434.9pt;left:187.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup></p> <p style="top:448.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rekurzió határozza meg. Az első néhány függvény</span></p> <p style="top:468.9pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></p> <p style="top:494.8pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:503.0pt;left:197.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:524.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:532.8pt;left:223.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:554.3pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:562.5pt;left:223.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:562.5pt;left:248.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">24</span></p> <p style="top:584.1pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:592.3pt;left:223.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:592.3pt;left:248.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup></p> <p style="top:592.3pt;left:277.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">120</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:608.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ebből sejthetjük, hogy</span></p> <p style="top:638.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></p> <p style="top:629.0pt;left:214.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:630.4pt;left:209.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:653.9pt;left:209.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:630.7pt;left:227.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup></p> <p style="top:647.0pt;left:228.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:630.7pt;left:262.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup></p> <p style="top:647.0pt;left:253.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)!</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:668.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Valóban:</span></p> <p style="top:696.0pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span></p> <p style="top:683.5pt;left:128.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:707.7pt;left:134.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1 +</span></sup></p> <p style="top:683.5pt;left:248.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:707.7pt;left:254.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span></sup></p> <p style="top:686.2pt;left:310.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:687.6pt;left:305.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:711.1pt;left:306.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:687.9pt;left:323.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup></p> <p style="top:704.2pt;left:324.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:687.9pt;left:357.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup></p> <p style="top:704.2pt;left:349.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)!</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dξ</span></i></sup></p> <p style="top:732.3pt;left:215.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></p> <p style="top:722.6pt;left:282.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:724.0pt;left:277.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:747.4pt;left:278.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:724.2pt;left:303.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup></p> <p style="top:740.5pt;left:295.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:724.2pt;left:356.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span></sup></p> <p style="top:740.5pt;left:348.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)!</span></p> <p style="top:768.7pt;left:215.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></p> <p style="top:758.9pt;left:277.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></p> <p style="top:760.3pt;left:278.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:783.8pt;left:278.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:760.6pt;left:296.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup></p> <p style="top:776.9pt;left:297.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:760.6pt;left:331.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span></sup></p> <p style="top:776.9pt;left:323.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_273.png	"További gyakorló feladatok 6. Számoljuk ki az y(z) — "2 dííerenci három közelítő függvényt, ha a kezdeti feltétel y(l) — 1. . Megoldás. A kezdeti függvény konstans v(z) — 1. a továbbiakat módon definiáljuk. Ebből ernlr e éo 1l éo1 e ; EX (azf , (r éc 14 , BZ , 2a adódik, ami alapján rájöhetűnk, hogy e) ennek limesze (r 2. 0 esetén) y(z) 1. ami megoldja a kezdetiérté 7. Szukcesszív approximáció se tet v(0) — 1 kozdet AMegoldás. A tüs ealH 15 64 edoyas er -1 vel határozzuk meg az y(z) — 7-4-y(r) diff feltételhez tartozó megoldását. tvénysorozat 0. tagja olz) — 1. a továbbiakat a e eddelirtőtő edrelárttőtői e oltrté. ebből sejthetjük, hogy Valóban: 14 [é edO ME 14 mrE E el). dlet szükcesszív approximációjával kapott első
A függvénysorozat mindenhol abszolút konvergens, határértéke xi y(x) = lim k→∞−1 −x + 2 y(x) = lim k→ i=0 xi i! + xk+1 (k + 1 x + (k + 1)! = −1 −x + 2ex Ez megoldja az egyenletet: (−1 −x + 2ex)′ = −1 + 2ex = x + (−1 −x + 2ex). 8. Határozzuk meg a (2x + 1)y′ −3y = 0 diﬀerenciálegyenlet általános megoldását. Megoldás. Az egyenlet y′ y = 3 2x + 1 alakra hozható, ha y ̸= 0 és x ̸= −1 2 (ha y valahol 0, akkor mindenhol az az egyenlet alapján). Integráljuk mindkét oldalt felhasználva, hogy y és y0 egyező előjelű: ln y y Z x y′(ξ) Z x y (ξ) y(ξ) dξ = 2 3 2ξ + 1dξ = 3 3 2 ln 1 + 2x 1 + 2x0 = ln y0 x0 y(ξ) dξ = x0 2ξ + 1dξ = 2 ln 1 + 2x0 azaz y0 x0 x0 y(x) = y0 1 + 2x 3/2 1 + 2x0 Az értelmezési tartomány (−∞, −1 2 1 2) vagy (−1 2 h h j k Az értelmezési tartomány (−∞, −1 2) vagy (−1 2, ∞). Látszólag két szabad paraméter van, de az előjelre ügyelve a nevezőt kihozhatjuk a zárójelből, és y0-val összevonhatjuk: y(x) = C\|1 + 2x\|3/2, ahol C bármilyen előjelű lehet. 9. Határozzuk meg az (1 + x2)y′ + (1 + y2) = 0 diﬀerenciálegyenlet általános megoldását. Megoldás. Az egyenlet szétválasztható: y′ 1 + y2 = −1 1 + x2. Mivel nincsen megadva kezdeti feltétel, egyszerűbb határozatlan integrállal folytatni, ügyelve arra, hogy a két oldal egy-egy primitív függvénye konstansban eltérhet egymástól: arctan y(x) = C −arctan x tehát y(x) = tan(C −arctan x). Egyszerűbb alakot kaphatunk tan(α ± β) = tan α ± tan β 1 ∓tan α tan β felhasználásával: y(x) = tan(C −arctan x) = tan C −x 1 + (tan C)x vagy tan C helyett C-t írva (ez is tetszőleges konstans!): y(x) = C −x 1 + Cx.	A függvénysorozat mindenhol abszolút konvergens, határértéke _y(x) = lim_ _k→∞_ _[−][1][ −]_ _[x][ + 2]_ Ez megoldja az egyenletet: _k_ � _i=0_ _x[i]_ _x[k][+1]_ _i! [+]_ (k + 1)! [=][ −][1][ −] _[x][ + 2][e][x]_ (−1 − _x + 2e[x])[′]_ = −1 + 2e[x] = x + (−1 − _x + 2e[x])._ 8. Határozzuk meg a (2x + 1)y[′] _−_ 3y = 0 differenciálegyenlet általános megoldását. _Megoldás. Az egyenlet_ _y[′]_ 3 _y_ [=] 2x + 1 alakra hozható, ha y ̸= 0 és x ̸= − [1] 2 [(ha][ y][ valahol 0, akkor mindenhol az az egyenlet] alapján). Integráljuk mindkét oldalt felhasználva, hogy y és y0 egyező előjelű: � _x_ _y[′](ξ)_ � _x_ 3 ln _[y]_ = _y0_ _x0_ _y(ξ) [d][ξ][ =]_ _x0_ 2ξ + 1[dξ][ = 3]2 [ln 1 + 2]1 + 2x[x]0 azaz 3/2 � _y(x) = y0_ � 1 + 2x 1 + 2x0 Az értelmezési tartomány (−∞, − [1] 2 [) vagy (][−] [1]2 _[,][ ∞][). Látszólag két szabad paraméter van,]_ de az előjelre ügyelve a nevezőt kihozhatjuk a zárójelből, és y0-val összevonhatjuk: _y(x) = C\|1 + 2x\|[3][/][2],_ ahol C bármilyen előjelű lehet. 9. Határozzuk meg az (1 + x[2])y[′] + (1 + y[2]) = 0 differenciálegyenlet általános megoldását. _Megoldás. Az egyenlet szétválasztható:_ _y[′]_ _−1_ 1 + y[2][ =] 1 + x[2] _[.]_ Mivel nincsen megadva kezdeti feltétel, egyszerűbb határozatlan integrállal folytatni, ügyelve arra, hogy a két oldal egy-egy primitív függvénye konstansban eltérhet egymástól: arctan y(x) = C − arctan x tehát y(x) = tan(C − arctan x). Egyszerűbb alakot kaphatunk tan(α ± β) = [tan][ α][ ±][ tan][ β] 1 ∓ tan α tan β felhasználásával: tan C − _x_ _y(x) = tan(C −_ arctan x) = 1 + (tan C)x vagy tan C helyett C-t írva (ez is tetszőleges konstans!): _y(x) =_ _[C][ −]_ _[x]_ 1 + Cx[.] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A függvénysorozat mindenhol abszolút konvergens, határértéke</span></p> <p style="top:92.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = lim</span></p> <p style="top:102.6pt;left:144.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">→∞</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup></p> <p style="top:82.5pt;left:231.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:83.9pt;left:226.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:107.3pt;left:226.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:84.1pt;left:243.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup></p> <p style="top:100.4pt;left:245.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:84.2pt;left:279.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup></p> <p style="top:100.4pt;left:270.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)! </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:124.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez megoldja az egyenletet:</span></p> <p style="top:149.5pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:174.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:193.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet</span></p> <p style="top:215.5pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:231.8pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:215.5pt;left:148.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:231.8pt;left:134.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></p> <p style="top:255.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakra hozható, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:262.6pt;left:271.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ha</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valahol</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor mindenhol az az egyenlet</span></sup></p> <p style="top:269.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alapján). Integráljuk mindkét oldalt felhasználva, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyező előjelű:</span></p> <p style="top:301.1pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup></p> <p style="top:309.3pt;left:119.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:301.1pt;left:134.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:288.7pt;left:146.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:312.9pt;left:152.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span></p> <p style="top:293.0pt;left:165.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:309.3pt;left:166.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:288.7pt;left:220.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:312.9pt;left:225.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span></p> <p style="top:293.0pt;left:251.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:309.3pt;left:238.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">dξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span></sup></p> <p style="top:309.3pt;left:300.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln 1 + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:309.3pt;left:322.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:328.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span></p> <p style="top:358.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:345.7pt;left:156.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:366.8pt;left:165.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:345.7pt;left:203.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:389.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az értelmezési tartomány</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−∞</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:397.1pt;left:255.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vagy</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:397.1pt;left:312.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∞</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Látszólag két szabad paraméter van,</span></sup></p> <p style="top:404.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">de az előjelre ügyelve a nevezőt kihozhatjuk a zárójelből, és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-val összevonhatjuk:</span></p> <p style="top:429.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:453.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> bármilyen előjelű lehet.</span></p> <p style="top:472.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Határozzuk meg az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:491.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet szétválasztható:</span></p> <p style="top:513.7pt;left:118.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:530.0pt;left:107.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:513.7pt;left:165.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:530.0pt;left:156.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:551.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mivel nincsen megadva kezdeti feltétel, egyszerűbb határozatlan integrállal folytatni, ügyel-</span></p> <p style="top:566.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ve arra, hogy a két oldal egy-egy primitív függvénye konstansban eltérhet egymástól:</span></p> <p style="top:591.2pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arctan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arctan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:616.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = tan(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arctan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Egyszerűbb alakot kaphatunk</span></p> <p style="top:646.7pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tan(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> β</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tan</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tan</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> β</span></i></sup></p> <p style="top:654.9pt;left:178.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∓</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> β</span></i></p> <p style="top:676.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felhasználásával:</span></p> <p style="top:704.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = tan(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">arctan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:696.3pt;left:259.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:712.6pt;left:252.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + (tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:735.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vagy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyett</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-t írva (ez is tetszőleges konstans!):</span></p> <p style="top:766.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:774.7pt;left:145.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_274.png	A függvénysorozat mindenhol abszolút konvergens, határértéke véa) — lm -1 1—rt2s Ez megoldja az egy: [1 2ÉY--1427 r4(-1-—r420). Határozzuk meg a (2r 4. 14 — 3y — 0 dilferenciálegyenlet általános megoldását. .Megoldás. Az egyenlet v y 21 alakra hozható, ha y 3 0 és 2 2—! (ha y valahol 0, akkor mindenhol az az egyenlet hh er gti á (—00,-—1) vagy (-1.00). Látszólag két szabad paraméter van, de az előjelre ügyelve a nevezőt kihozhatjuk a zárójelből, és gy-val összevonhatjuk: w — C 4-22 9. ahol C bármilyen előjelű lehet. Határozzuk meg az (14-29jy 4 (14- ) — 0 differenciálegyenlet általános megoldását. .Megoldás. Az egyenlet szétválasztható: y 1 14 137 Mivel nincsen megadva kezdeti feltétel, egyszerűbb határozatlan integrállal folytatni, ügyel- ve arra, hogy a két oldal egy-egy primitív fűggvénye konstansban eltérhet egyi zástól: aretany(z) aretanz tehát y(z) — tan(C — aretan ). Egyszerűbb alakot kaphatunk tana ktanő tanla 2)— Tetanotanő felhaszmálásával: , FK a) — tan(C — aretána) — OJ vagy tan C helyett C-t írva (ez is tetszőleges konstans!) C-s MlNYET 7
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 8. feladatsor: Kezdeti feltételtől való függés, egzakt diﬀerenciálegyenletek 1. Keressük meg az y′ = sin y diﬀerenciálegyenlet konstans megoldásait, és határozzuk meg ezek kezdeti feltétel szerinti deriváltjait, ha a kezdeti feltétel az x0 = 0 pontban van megadva. 2. Tekintsük az y′ 1 y′ 1 = y2 y′ 2 = − y′ 2 = −sin y1 diﬀerenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. 3. Oldjuk meg a 2x + cos y −(x sin y)y′ = 0 diﬀerenciálegyenletet y(1) = 0 kezdeti feltétel mellett. 4. Egyváltozós multiplikátorral tegyük egzakttá az x3 +y4 +8xy3y′ = 0 diﬀerenciálegyenletet, majd oldjuk meg. 5. Egyváltozós multiplikátorral tegyük egzakttá az y ln y + y sinh x + (x + yey)y′ = 0 diﬀeren- ciálegyenletet, majd oldjuk meg. További gyakorló feladatok 6. Határozzuk meg az y′ = sin(xy), y(0) = 0 kezdetiérték-probléma megoldásának kezdeti feltétel szerinti deriváltját. 7. Tekintsük az y′ 1 y′ 1 = xe−y1y2 y′ 2 = 1 −ey2 y′ 2 = 1 −ey2 diﬀerenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. 8. Határozzuk meg az y cosh x + (sinh x −2y)y′ = 0 diﬀerenciálegyenlet y(0) = 1 kezdeti feltételnek eleget tevő megoldását. 9. Oldjuk meg az y cosh x+(sinh x−2y)y′ = 0 diﬀerenciálegyenletet y(0) = −1 kezdeti feltétel mellett. 10. Határozzuk meg az x+y y 10. Határozzuk meg az x+y y + 2x+3y2 2y y′ = 0 diﬀerenciálegyenlet általános megoldását. 11. Oldjuk meg az y + (yex −1)y′ = 0 diﬀerenciálegyenletet y(0) = 3 kezdeti feltétel mellett. + 2x+3y2 2y	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 8. feladatsor: Kezdeti feltételtől való függés, egzakt differenciálegyenletek 1. Keressük meg az y[′] = sin y differenciálegyenlet konstans megoldásait, és határozzuk meg ezek kezdeti feltétel szerinti deriváltjait, ha a kezdeti feltétel az x0 = 0 pontban van megadva. 2. Tekintsük az _y1[′]_ [=][ y][2] _y2[′]_ [=][ −] [sin][ y][1] differenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. 3. Oldjuk meg a 2x + cos y − (x sin y)y[′] = 0 differenciálegyenletet y(1) = 0 kezdeti feltétel mellett. 4. Egyváltozós multiplikátorral tegyük egzakttá az x[3] + _y[4]_ +8xy[3]y[′] = 0 differenciálegyenletet, majd oldjuk meg. 5. Egyváltozós multiplikátorral tegyük egzakttá az y ln y + y sinh x + (x + ye[y])y[′] = 0 differenciálegyenletet, majd oldjuk meg. ## További gyakorló feladatok 6. Határozzuk meg az y[′] = sin(xy), y(0) = 0 kezdetiérték-probléma megoldásának kezdeti feltétel szerinti deriváltját. 7. Tekintsük az _y1[′]_ [=][ xe][−][y][1][y][2] _y2[′]_ [= 1][ −] _[e][y][2]_ differenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. 8. Határozzuk meg az y cosh x + (sinh x − 2y)y[′] = 0 differenciálegyenlet y(0) = 1 kezdeti feltételnek eleget tevő megoldását. 9. Oldjuk meg az y cosh x+(sinh x−2y)y[′] = 0 differenciálegyenletet y(0) = −1 kezdeti feltétel mellett. 10. Határozzuk meg az _[x][+][y]_ + [2][x][+3][y][2] _y[′]_ = 0 differenciálegyenlet általános megoldását. _y_ 2y 11. Oldjuk meg az y + (ye[x] _−_ 1)y[′] = 0 differenciálegyenletet y(0) = 3 kezdeti feltétel mellett. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">8. feladatsor: Kezdeti feltételtől való függés, egzakt</span></b></p> <p style="top:106.3pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">diﬀerenciálegyenletek</span></b></p> <p style="top:146.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Keressük meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet konstans megoldásait, és határozzuk meg</span></p> <p style="top:160.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezek kezdeti feltétel szerinti deriváltjait, ha a kezdeti feltétel az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban van meg-</span></p> <p style="top:175.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adva.</span></p> <p style="top:191.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Tekintsük az</span></p> <p style="top:218.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:224.2pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:235.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:241.6pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:261.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását. Ha-</span></p> <p style="top:276.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tározzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban.</span></p> <p style="top:292.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Oldjuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:307.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:323.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Egyváltozós multiplikátorral tegyük egzakttá az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet,</span></p> <p style="top:338.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">majd oldjuk meg.</span></p> <p style="top:354.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Egyváltozós multiplikátorral tegyük egzakttá az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ye</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀeren-</span></p> <p style="top:369.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciálegyenletet, majd oldjuk meg.</span></p> <p style="top:401.7pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:425.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdetiérték-probléma megoldásának kezdeti</span></p> <p style="top:440.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">feltétel szerinti deriváltját.</span></p> <p style="top:456.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Tekintsük az</span></p> <p style="top:483.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:489.2pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:500.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:506.7pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:526.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását. Ha-</span></p> <p style="top:541.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tározzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban.</span></p> <p style="top:557.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti</span></p> <p style="top:572.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">feltételnek eleget tevő megoldását.</span></p> <p style="top:588.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:603.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:619.6pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Határozzuk meg az</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup></p> <p style="top:626.9pt;left:186.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:619.6pt;left:200.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:626.9pt;left:223.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:619.6pt;left:242.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:637.2pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ye</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> </div>	page_275.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 8. feladatsor: Kezdeti feltételtő differenciálegyenletek való függés, egzakt 1. Keressük meg az y — siny dilferenciálegyenlet konstans megoldásait, és határozzuk meg cezek kezdeti feltétel szerinti deríváltjait, ha a kezdeti feltétel az 9 — 0 pontban van me adva. 2. Tekintsük az űn e —sínyi dífferenciálegyenlet-rendszer (0) — ye(0) — 0 kezdeti feltételt kielégítő megoldását. Ha- tározzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. 3. Oldjuk meg a 2r 4 cosy — (rsinyjy — 0 ditferenciálegyenletet y(1) — 0 kezdeti feltétel sellett. 4. Egyváltozós multiplikátorral tegyük egzakttá az 2 4-4-4-Szyév — 0 differenciálegyenletet majd oldjuk meg. 5. Egyváltozós multiplikátorral tegyűk egzakttá az ylny 4-ysinh r 4 (r 4 yey — 0 dítleret clálegyenletet, majd oldjuk meg. "További gyakorló feladatok 6. Határozzuk meg az 4 — sin(zy), v(0) — 0 kezdetiérték-problóma megoldásának kezdeti feltétel szerinti deriváltját. 7. Tekintsük az úr z1-et differenciálegyenlet-rendszer y,(0) — y4(0) — 0 kezdeti feltételt kielégítő megoldását. Ha- 8. Határozzuk meg az ycoshz 4. (sinhz — 2y)y — 0 dilerenciálegyenlet y(0) 9. Oldjuk mog az yeoshz4 (sinh 1— 2997 mellett kezdeti diflerenciálogyenletet y(0) — —1 kezdeti feltétel 11. Oldjuk meg az y 4. (ye" — 1w — 0 differenciálegyenletet y(0) — 3 kezdeti feltétel mellett. 10. Határozzuk meg az w — 0 diflerenciálegyenlet általános megoldását
Megoldás. Létezik csak y-tól függő multiplikátor: ∂2x+3y2 2y ∂( x+y y ) ∂( x+y y Z 1 y dy = ln y + C, ln \|M(y)\| = 2y ∂x − ∂y x+y dy = Z 1 tehát M(y) = y, és így x + y + (x + 3 2y2)y′ egzakt. Egy potenciál 3 2y2)y′ egzakt. Egy potenciál Z x 0 (ξ + 0) dξ + Z x Z y dη = x2 2 u(x, y) = x + 3 2η2 x + 3 2 x2 2 + xy + y3 2 y 2 . Az általános megoldás implicit alakban u(x, y(x)) = C, ahol C paraméter. 11. Oldjuk meg az y + (yex −1)y′ = 0 diﬀerenciálegyenletet y(0) = 3 kezdeti feltétel mellett. Megoldás. Létezik csak x-től függő multiplikátor: ln \|M(x)\| = ∂y ∂y −∂(yex−1) ∂x yex −1 dx = (−1) dx = −x, tehát M(x) = e−x és ye−x + (y −e−x)y′ = 0 egzakt. Egy potenciál Z x 0 0 · e−ξ dξ + Z y u(x, y) = Z y 2 0 (η −e−x) dη = y2 y 2 −ye−x. A kezdeti feltétel alapján a megoldás implicit alakja u(x, y(x)) = u(x0, y0) = 3 2. Az explicit alakot is meg lehet határozni: A kezdeti feltétel alapján a megoldás implicit alakja u(x, y(x)) = u(x0, y0) = 3 2 1 + 3e2x. y(x) = e−x + e−x√ y(x) = e−x + e−x√	_Megoldás. Létezik csak y-tól függő multiplikátor:_ _∂_ [2][x][+3]2y _[y][2]_ _∂(_ _[x][+]y_ _[y]_ [)] ln \|M (y)\| = � _∂x_ _x+−y_ _∂y_ dy = � 1 _y_ [d][y][ = ln][ y][ +][ C,] _y_ tehát M (y) = y, és így x + y + (x + [3] 2 _[y][2][)][y][′][ egzakt. Egy potenciál]_ � _x_ � _y_ _u(x, y) =_ 0 [(][ξ][ + 0) d][ξ][ +] 0 � � _x + [3]_ dη = _[x][2]_ 2[η][2] 2 [+][ xy][ +][ y]2[3] _[.]_ Az általános megoldás implicit alakban u(x, y(x)) = C, ahol C paraméter. 11. Oldjuk meg az y + (ye[x] _−_ 1)y[′] = 0 differenciálegyenletet y(0) = 3 kezdeti feltétel mellett. _Megoldás. Létezik csak x-től függő multiplikátor:_ _∂y_ � _∂y_ _[−]_ _[∂][(][ye]∂x[x][−][1)]_ � ln \|M (x)\| = dx = (−1) dx = −x, _ye[x]_ _−_ 1 tehát M (x) = e[−][x] és ye[−][x] + (y − _e[−][x])y[′]_ = 0 egzakt. Egy potenciál � _x_ � _y_ _u(x, y) =_ 0 [0][ ·][ e][−][ξ][ d][ξ][ +] 0 [(][η][ −] _[e][−][x][) d][η][ =][ y]2[2]_ _[−]_ _[ye][−][x][.]_ A kezdeti feltétel alapján a megoldás implicit alakja u(x, y(x)) = u(x0, y0) = 2[3] [. Az explicit] alakot is meg lehet határozni: _y(x) = e[−][x]_ + e[−][x][√] 1 + 3e[2][x]. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Létezik csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-tól függő multiplikátor:</span></p> <p style="top:102.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:89.9pt;left:168.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:87.7pt;left:182.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">+3</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:92.4pt;left:197.7pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></p> <p style="top:99.7pt;left:194.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:92.3pt;left:219.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:87.7pt;left:232.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></sup></p> <p style="top:92.4pt;left:246.6pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:99.7pt;left:241.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:108.9pt;left:213.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:118.1pt;left:219.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:102.4pt;left:264.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:89.9pt;left:292.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></p> <p style="top:110.6pt;left:305.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C,</span></i></sup></p> <p style="top:139.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, és így</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:146.6pt;left:265.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egzakt. Egy potenciál</span></sup></p> <p style="top:173.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:161.5pt;left:156.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:185.7pt;left:161.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 0) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:161.5pt;left:234.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:185.7pt;left:240.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:161.0pt;left:251.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:173.9pt;left:259.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup></p> <p style="top:182.1pt;left:281.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:161.0pt;left:299.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:173.9pt;left:308.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:182.1pt;left:341.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:182.1pt;left:396.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:204.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az általános megoldás implicit alakban</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméter.</span></p> <p style="top:224.2pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ye</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:243.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Létezik csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től függő multiplikátor:</span></p> <p style="top:280.6pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:268.1pt;left:169.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:268.6pt;left:183.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:277.8pt;left:183.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ye</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup></p> <p style="top:277.8pt;left:222.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:288.8pt;left:195.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ye</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:280.6pt;left:249.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:268.1pt;left:278.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:280.6pt;left:288.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></p> <p style="top:313.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ye</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egzakt. Egy potenciál</span></p> <p style="top:346.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:334.2pt;left:156.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:358.5pt;left:161.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:334.2pt;left:232.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:358.5pt;left:238.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:354.9pt;left:329.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ye</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:378.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltétel alapján a megoldás implicit alakja</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:385.4pt;left:469.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az explicit</span></sup></p> <p style="top:392.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakot is meg lehet határozni:</span></p> <p style="top:418.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup></p> <p style="top:418.9pt;left:203.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_276.png	1. Megoldás. Létezik csak y-tól függő multiplikátor: E" 9N jar f aa flaízíy té o] EE 492 fjdv l9 tehát M(g) — 4. és így - y 4. (z 4. yjy eszakt. Egy potenciól vlz2 er 0464 f(es a Za £ Az általános megoldás implicit alakban ulr, y(z)) — C, ahol C param 4 y(0) — 3 kezdeti feltétel mellett. Oldjuk meg az y 4. (ye? — 19 — 0 dílferenciálogyenle .Megoldás. Létezik csak 2. v ől függő multiplikátor: maro- / tehát M(r! as- fevas syet YE v f9-eaés [ A kezdeti feltétel alapján a megoldás implicit alakja u(z, y(2)) — ulro. ) — 3. Az explicit LORTSAT M ERZEI egzakt. Egy potenciál ja -
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 8. feladatsor: Kezdeti feltételtől való függés, egzakt diﬀerenciálegyenletek (megoldás) 1. Keressük meg az y′ = sin y diﬀerenciálegyenlet konstans megoldásait, és határozzuk meg ezek kezdeti feltétel szerinti deriváltjait, ha a kezdeti feltétel az x0 = 0 pontban van megadva. Megoldás. Ha y(x) = y0 konstans megoldás, akkor y′(x) = 0, tehát sin y0 = 0. Ez y0 = kπ esetén teljesül (k ∈Z). Rögzítsünk egy k egész számot, és jelölje Y (x, 0, y0) az y(x0) = y0 kezdeti feletételt kielégítő megoldást, ennek y0 szerinti deriváltja az (x, 0, kπ) pontban legyen D(x). A tanult tétel szerint Y és D diﬀerenciálható, a láncszabály alapján D′(x) = ∂ ∂y0 ∂ ∂xY (x, 0, y0) y0=kπ ∂y0 sin Y (x, 0, y0) ∂y0 y0=kπ = cos Y (x, 0, kπ)D(x) = (−1)kD(x). A kapott diﬀerenciálegyenlet szétválasztható: D′(x) D(x) = (−1)k, a két oldalt integrálva D(0) = 1 ﬁgyelembevételével ln D(x) = (−1)kx, vagyis D(x) = e(−1)kx adódik. 2. Tekintsük az y′ 1 y′ 1 = y2 y′ 2 = − y′ 2 = −sin y1 diﬀerenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. Megoldás. Az y1(x) = y2(x) = 0 konstans függvény a kezdetiérték-probléma egyetlen megoldása. Jelölje Y(x, 0, y0) = (Y1(x, 0, y0), Y2(x, 0, y0)) a megoldást (y1(0), y2(0)) = y0 = (y0,1, y0,2) kezdeti feltétel mellett. Legyen Dij(x) = ∂Yi(x,0,y0) ∂y0,j . A tanult tétel alapján Dij és Y diﬀerenciálható. f1(x, y) = y2, f2(x, y) = −sin y1 jelöléssel a derivált D′ ij D′ ij(x) = ∂ ∂y0,j ∂ ∂xYi(x, 0, y0) ∂y0,j fi(x, 0, Yi(x, 0, y0)) ∂y0,j ∂x y0=0 ∂y0,j y0=0 D′ 11(x) = D21(x) y0=0 D′ 11(x) = D21(x) D′ 12(x) = D22(x) D′ 12(x) = D22(x) D′ 21(x) = −D11(x D′ 21(x) = −D11(x) D′ 22(x) = −D12(x) ′ 22(x) = −D12(x).	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 8. feladatsor: Kezdeti feltételtől való függés, egzakt differenciálegyenletek (megoldás) 1. Keressük meg az y[′] = sin y differenciálegyenlet konstans megoldásait, és határozzuk meg ezek kezdeti feltétel szerinti deriváltjait, ha a kezdeti feltétel az x0 = 0 pontban van megadva. _Megoldás. Ha y(x) = y0 konstans megoldás, akkor y[′](x) = 0, tehát sin y0 = 0. Ez y0 = kπ_ esetén teljesül (k ∈ Z). Rögzítsünk egy k egész számot, és jelölje Y (x, 0, y0) az y(x0) = _y0 kezdeti feletételt kielégítő megoldást, ennek y0 szerinti deriváltja az (x, 0, kπ) pontban_ legyen D(x). A tanult tétel szerint Y és D differenciálható, a láncszabály alapján _∂_ _D[′](x) =_ _∂y0_ _∂_ _∂x_ _[Y][ (][x,][ 0][, y][0][)]_ ��y0=kπ _∂_ = sin Y (x, 0, y0) _∂y0_ ��y0=kπ = cos Y (x, 0, kπ)D(x) = (−1)[k]D(x). A kapott differenciálegyenlet szétválasztható: _D[′](x)_ _D(x) [= (][−][1)][k][,]_ a két oldalt integrálva D(0) = 1 figyelembevételével ln D(x) = (−1)[k]x, vagyis _D(x) = e[(][−][1)][k][x]_ adódik. 2. Tekintsük az _y1[′]_ [=][ y][2] _y2[′]_ [=][ −] [sin][ y][1] differenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. _Megoldás. Az y1(x) = y2(x) = 0 konstans függvény a kezdetiérték-probléma egyetlen meg-_ oldása. Jelölje Y(x, 0, y0) = (Y1(x, 0, y0), Y2(x, 0, y0)) a megoldást (y1(0), y2(0)) = y0 = (y0,1, y0,2) kezdeti feltétel mellett. Legyen Dij(x) = _[∂Y][i]∂y[(][x,]0[0],j[,][y][0][)]_ . A tanult tétel alapján Dij és Y differenciálható. f1(x, y) = y2, f2(x, y) = − sin y1 jelöléssel a derivált _∂_ _∂_ _∂_ _Dij[′]_ [(][x][) =] = _fi(x, 0, Yi(x, 0, y0))_ _∂y0,j_ _∂x_ _[Y][i][(][x,][ 0][,][ y][0][)]��y0=0_ _∂y0,j_ ��y0=0 _D11[′]_ [(][x][) =][ D][21][(][x][)] _D12[′]_ [(][x][) =][ D][22][(][x][)] _D21[′]_ [(][x][) =][ −][D][11][(][x][)] _D22[′]_ [(][x][) =][ −][D][12][(][x][)][.] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">8. feladatsor: Kezdeti feltételtől való függés, egzakt</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">diﬀerenciálegyenletek (megoldás)</span></b></p> <p style="top:146.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Keressük meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet konstans megoldásait, és határozzuk meg</span></p> <p style="top:161.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezek kezdeti feltétel szerinti deriváltjait, ha a kezdeti feltétel az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban van meg-</span></p> <p style="top:175.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adva.</span></p> <p style="top:194.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> konstans megoldás, akkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ez</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> kπ</span></i></p> <p style="top:209.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">esetén teljesül (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">Z</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">). Rögzítsünk egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egész számot, és jelölje</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:223.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feletételt kielégítő megoldást, ennek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szerinti deriváltja az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, kπ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban</span></p> <p style="top:237.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tanult tétel szerint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható, a láncszabály alapján</span></p> <p style="top:270.4pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:262.3pt;left:160.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:278.6pt;left:155.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:262.3pt;left:178.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:278.6pt;left:174.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:254.0pt;left:241.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:286.0pt;left:244.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kπ</span></i></p> <p style="top:306.3pt;left:140.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:298.2pt;left:160.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:314.5pt;left:155.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:306.3pt;left:175.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:290.0pt;left:244.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:321.9pt;left:247.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kπ</span></i></p> <p style="top:334.4pt;left:140.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, kπ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:353.2pt;left:140.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:377.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kapott diﬀerenciálegyenlet szétválasztható:</span></p> <p style="top:400.2pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:416.5pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:440.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a két oldalt integrálva</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ﬁgyelembevételével</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagyis</span></p> <p style="top:467.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:492.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik.</span></p> <p style="top:511.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Tekintsük az</span></p> <p style="top:536.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:542.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:553.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:559.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:578.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását. Ha-</span></p> <p style="top:593.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tározzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban.</span></p> <p style="top:612.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> konstans függvény a kezdetiérték-probléma egyetlen meg-</span></p> <p style="top:626.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">oldása. Jelölje</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a megoldást</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)) =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:641.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂Y</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">i</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x,</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">y</span></b></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:649.3pt;left:352.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">,j</span></i></p> <p style="top:641.9pt;left:384.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A tanult tétel alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></p> <p style="top:658.4pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálható.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelöléssel a derivált</span></p> <p style="top:690.9pt;left:108.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:697.0pt;left:117.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup></p> <p style="top:682.8pt;left:167.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:699.1pt;left:159.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,j</span></i></p> <p style="top:682.8pt;left:188.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:699.1pt;left:184.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:674.6pt;left:253.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:706.5pt;left:257.0pt;line-height:8.0pt"><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">y</span></b><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:690.9pt;left:281.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:682.8pt;left:304.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:699.1pt;left:295.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,j</span></i></p> <p style="top:690.9pt;left:320.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, Y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></p> <p style="top:674.6pt;left:415.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:706.5pt;left:419.2pt;line-height:8.0pt"><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">y</span></b><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:719.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:725.1pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:736.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:742.6pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:753.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:760.0pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:771.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:777.4pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> </div>	page_277.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 8. feladatsor: Kezdeti feltételtől való függés, egzakt differenciálegyenletek (megoldás) 1. Keressük meg az y — siny dilferenciálegyenlet konstans megoldásait, és határozzuk meg cezek kezdeti felt ltjait, ha a kezdeti feltétel az z9 — 0 pontban van meg. adva. el szerinti deri "Megoldás. Ha y(z) — y9 konstans megoldás, akkor y(2) — 0, tehát siny, — 0. Ez 4 esetén teljesül (k € Z). Röszítsünk egy k egész számot, és jelölje Y(.0,yo) az v(zo) Wo kezdeti feletételt kielégítő megoldást, ennek y szerinti deriváltja az (z.0, kr) pontban legyen Díz). A tanult tétel szerint Y és D diflerenciálható, a láncszabály alapján 0 0 D) öryöz 3 ú - ögAYE B ) osY(2,0.ke)Dz) 14D(-). A kapott differenciálegyenlet szétválasztható: D) 9. po a két oldalt integrálva D(0) — 1 figyelembevételével In Dír) — (—1)4r, vagyis. Día) — s adódik. 2. Tekintsük az úzn e —sínyi dífferenciálegyenlet-rendszer (0) — ye(0) — 0 kezdeti feltételt kielégítő megoldását. Ha- tározzuk meg a megoldás deriváltját a kezdeti a a pontban. A szeránt ebb oldása.. Jelölje Víz.0,yo) — (K(2.0yo). Y. O.ye)) a megoldást (y.(0).9(0)) — Yo (Uva, 902) kezdeti feltétel mellett. Legyen Dulz) — PÉSZSZA. A tanult tétel alapján D. és V differenciálható. filr.y) — 92. falr,y) — — sinyn jelöléssel a derivált 22 fisOXY(r.Oyo) DA pztlndyő] -7x Dyos 0r Dule) — Da(z) Dile) — Dod(2) Dale) — —Dulr) Díule) — —D.
Ez két független diﬀerenciálegyenlet-rendszer, a kezdeti feltétel D(0) = I (egységmátrix). Később látni fogjuk, hogy hogyan lehet az egyenletrendszert megoldani, de azt most is észrevehetjük, hogy D(x) = # " D11(x) D12(x) D21(x) D22(x) # cos x sin x −sin x cos x megoldás (azt tudjuk, hogy csak egy megoldás létezik). 3. Oldjuk meg a 2x + cos y −(x sin y)y′ = 0 diﬀerenciálegyenletet y(1) = 0 kezdeti feltétel mellett. Megoldás. ∂(2x + cos y) + cos y) ∂y = −sin y = ∂(−x sin y) ∂x ∂x miatt az egyenlet egzakt, a potenciál u(x, y) = Z x Z y Z 0 (2ξ + cos 0) dξ + Zy 0 (−x sin η) dη 0 0 = x + x2 −x + x cos y = x2 + x cos y. A kezdeti feltétel alapján u(x0, y0) = u(1, 0) = 2, tehát a megoldás implicit alakban x2 + x cos y(x) = 2. Észrevehetjük, hogy ez az implicit egyenlet olyan görbét határoz meg a síkon, ami nem egy függvény graﬁkonja, hiszen ha egy (x, y) pár rajta van a görbén, akkor (x, ±y + 2kπ) is. Ilyenkor általában az történik, hogy a kezdeti feltételtől elindulva a görbe egy darabja még függvény, de amint az érintő függőlegessé válik, már nem biztos, hogy folytatódik a megoldás. Mindenesetre ha az (x0, y0) pontban nem függőleges az érintő, akkor annak egy környezetében egyértelmű a megoldás. A feladatbeli kezdeti feltétel viszont éppen a kivételes eset, hiszen az egyenletbe helyettesítve 2 + 1 −0y′(1) = 0 adódik, aminek nincsen megoldása. Az implicit alakból y(x)-et kifejezve két olyan függvényt találunk, ami megoldja az egyenletet egy nyílt intervallumon, aminek egyik végpontja 1, oda folytonosan kiterjed, és kielégíti a kezdeti feltételt: y(x) = ± arccos 2 x −x 4. Egyváltozós multiplikátorral tegyük egzakttá az x3 +y4 +8xy3y′ = 0 diﬀerenciálegyenletet, majd oldjuk meg. Megoldás. Létezik csak x-től függő multiplikátor: ∂(x3+y4) 3+y4) ∂y −∂(8xy3) ∂x Z −1 ln \|M(x)\| = ∂x 8xy3 dx = −1 2 2x dx = −1 1 2 ln \|x\| + C alapján M(x) = \|x\|−1/2, tehát \|x\|−1/2(x3 + y4) + 8\|x\|−1/2xy3y′ = 0 egzakt. A potenciál Z x 0 \|ξ\|−1/2(ξ3 + 04) dξ + Z y u(x, y) = Z y 7 0 8\|x\|−1/2xη3 dη = 2 2 7x4\|x\|−1/2 + 2xy4\|x\|−1/2. Az általános megoldás u(x, y) = C, amiből q \|x\| y(x) = u tC q \|x\| 7 2x −x3 x 7 .	Ez két független differenciálegyenlet-rendszer, a kezdeti feltétel D(0) = I (egységmátrix). Később látni fogjuk, hogy hogyan lehet az egyenletrendszert megoldani, de azt most is észrevehetjük, hogy � cos x sin x _−_ sin x cos x � _D(x) =_ �D11(x) _D12(x)�_ = _D21(x)_ _D22(x)_ megoldás (azt tudjuk, hogy csak egy megoldás létezik). 3. Oldjuk meg a 2x + cos y − (x sin y)y[′] = 0 differenciálegyenletet y(1) = 0 kezdeti feltétel mellett. _Megoldás._ _∂(2x + cos y)_ = − sin y = _[∂][(][−][x][ sin][ y][)]_ _∂y_ _∂x_ miatt az egyenlet egzakt, a potenciál � _x_ � _y_ _u(x, y) =_ 0 [(2][ξ][ + cos 0) d][ξ][ +] 0 [(][−][x][ sin][ η][) d][η] = x + x[2] _−_ _x + x cos y = x[2]_ + x cos y. A kezdeti feltétel alapján u(x0, y0) = u(1, 0) = 2, tehát a megoldás implicit alakban x[2] + _x cos y(x) = 2._ Észrevehetjük, hogy ez az implicit egyenlet olyan görbét határoz meg a síkon, ami nem egy függvény grafikonja, hiszen ha egy (x, y) pár rajta van a görbén, akkor (x, ±y + 2kπ) is. Ilyenkor általában az történik, hogy a kezdeti feltételtől elindulva a görbe egy darabja még függvény, de amint az érintő függőlegessé válik, már nem biztos, hogy folytatódik a megoldás. Mindenesetre ha az (x0, y0) pontban nem függőleges az érintő, akkor annak egy környezetében egyértelmű a megoldás. A feladatbeli kezdeti feltétel viszont éppen a kivételes eset, hiszen az egyenletbe helyettesítve 2 + 1 − 0y[′](1) = 0 adódik, aminek nincsen megoldása. Az implicit alakból y(x)-et kifejezve két olyan függvényt találunk, ami megoldja az egyenletet egy nyílt intervallumon, aminek egyik végpontja 1, oda folytonosan kiterjed, és kielégíti a kezdeti feltételt: � 2 � _y(x) = ± arccos_ _x_ _[−]_ _[x]_ _._ 4. Egyváltozós multiplikátorral tegyük egzakttá az x[3] + _y[4]_ +8xy[3]y[′] = 0 differenciálegyenletet, majd oldjuk meg. _Megoldás. Létezik csak x-től függő multiplikátor:_ ln \|M (x)\| = � _∂(x[3]∂y+y[4])_ _−_ _[∂][(8]∂x[xy][3][)]_ dx = � _−1_ 8xy[3] 2x [d][x][ =][ −][1]2 [ln][ \|][x][\|][ +][ C] alapján M (x) = \|x\|[−][1][/][2], tehát \|x\|[−][1][/][2](x[3] + y[4]) + 8\|x\|[−][1][/][2]xy[3]y[′] = 0 egzakt. A potenciál � _x_ � _y_ _u(x, y) =_ 0 _[\|][ξ][\|][−][1][/][2][(][ξ][3][ + 0][4][) d][ξ][ +]_ 0 [8][\|][x][\|][−][1][/][2][xη][3][ d][η][ = 2]7[x][4][\|][x][\|][−][1][/][2][ + 2][xy][4][\|][x][\|][−][1][/][2][.] Az általános megoldás u(x, y) = C, amiből � _\|x\|_ 2x _[−]_ _[x]7[3]_ _[.]_ _y(x) =_ � ��4 �C -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez két független diﬀerenciálegyenlet-rendszer, a kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (egységmátrix).</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Később látni fogjuk, hogy hogyan lehet az egyenletrendszert megoldani, de azt most is</span></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">észrevehetjük, hogy</span></p> <p style="top:120.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:104.1pt;left:148.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:112.7pt;left:153.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:112.7pt;left:198.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:127.2pt;left:153.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:127.2pt;left:198.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:104.1pt;left:232.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:120.0pt;left:241.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:104.1pt;left:254.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:112.7pt;left:265.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:112.7pt;left:305.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:127.2pt;left:260.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:127.2pt;left:304.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:104.1pt;left:328.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:152.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás (azt tudjuk, hogy csak egy megoldás létezik).</span></p> <p style="top:171.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Oldjuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:186.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:205.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:228.1pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:244.4pt;left:134.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:236.2pt;left:178.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:244.4pt;left:263.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:266.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">miatt az egyenlet egzakt, a potenciál</span></p> <p style="top:295.2pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:282.8pt;left:158.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:307.0pt;left:163.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos 0) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:282.8pt;left:260.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:307.0pt;left:265.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup></p> <p style="top:317.9pt;left:145.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y.</span></i></p> <p style="top:343.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltétel alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0) = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát a megoldás implicit alakban</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:358.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:372.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Észrevehetjük, hogy ez az implicit egyenlet olyan görbét határoz meg a síkon, ami nem</span></p> <p style="top:386.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egy függvény graﬁkonja, hiszen ha egy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pár rajta van a görbén, akkor</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">kπ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:401.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">is. Ilyenkor általában az történik, hogy a kezdeti feltételtől elindulva a görbe egy darabja</span></p> <p style="top:415.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">még függvény, de amint az érintő függőlegessé válik, már nem biztos, hogy folytatódik a</span></p> <p style="top:430.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás. Mindenesetre ha az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban nem függőleges az érintő, akkor annak egy</span></p> <p style="top:444.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">környezetében egyértelmű a megoldás.</span></p> <p style="top:459.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A feladatbeli kezdeti feltétel viszont éppen a kivételes eset, hiszen az egyenletbe helyette-</span></p> <p style="top:473.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sítve</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2 + 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adódik, aminek nincsen megoldása. Az implicit alakból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-et</span></p> <p style="top:488.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kifejezve két olyan függvényt találunk, ami megoldja az egyenletet egy nyílt intervallumon,</span></p> <p style="top:502.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">aminek egyik végpontja</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, oda folytonosan kiterjed, és kielégíti a kezdeti feltételt:</span></p> <p style="top:532.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> arccos</span></p> <p style="top:519.2pt;left:188.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:540.4pt;left:197.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:519.2pt;left:226.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:532.2pt;left:235.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:562.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Egyváltozós multiplikátorral tegyük egzakttá az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet,</span></p> <p style="top:576.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">majd oldjuk meg.</span></p> <p style="top:595.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Létezik csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től függő multiplikátor:</span></p> <p style="top:632.2pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:619.7pt;left:169.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:619.8pt;left:183.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">4</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:629.5pt;left:196.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:622.1pt;left:222.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(8</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">xy</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:629.5pt;left:245.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:640.4pt;left:212.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:632.2pt;left:269.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:619.7pt;left:298.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:640.4pt;left:312.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:640.4pt;left:369.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup></p> <p style="top:664.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 8</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egzakt. A potenciál</span></p> <p style="top:695.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:682.8pt;left:156.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:707.0pt;left:161.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 0</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:682.8pt;left:278.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:707.0pt;left:284.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xη</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span></sup></p> <p style="top:703.4pt;left:383.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:724.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span></p> <p style="top:766.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:760.6pt;left:147.4pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">4</span></p> <p style="top:741.8pt;left:144.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">v</span></p> <p style="top:747.3pt;left:144.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">u</span></p> <p style="top:753.3pt;left:144.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">u</span></p> <p style="top:759.3pt;left:144.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">t</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></p> <p style="top:745.7pt;left:165.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:756.1pt;left:175.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i></p> <p style="top:774.2pt;left:170.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:774.2pt;left:208.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_278.png	-rendszer, a kezdeti felté tlen ditferenciálogy el D(0) — 1 (egységmátrix). AKésőbb látni fogjuk, hogy hogyan lehet az egyenletrendszert megoldani, de azt most is észrevehetjűk, hogy 260 [s6) é]: E ] megoldás (azt tudjuk, hogy csak egy megoldás létezilő. iálegyenletet y(1) — 0 kezdeti feltétel Oldjuk meg a 2r 4. cosy — (rsinyjy — 0 díffere mellett. . Megoldás. dl2rtesy —a, Br 2] öz miatt az egyenlet egzakt, a potenciál Wa f/RE t cs0dé s [/Cssuyás mrtArárosy—tzosy, A kezdeti feltétel alapján ulzy. ) zeosylr) — 2. egy függvény grafikonja, hiszen ha egy (2.) pár rajta van a görbén, akkor (r,-Ey 4 2£7) A feladatbeli kezdeti feltétel viszont éppen a kívételes eset, hiszen az egyenletbe helyette- u(1,0) — 2. tehát a megoldás implicit alakban 2? 4. Egyváltozós multiplikátorral tegyük egzakttá az 2? 44 4-Szyé — 0 dífferenciálegyenletet "Megoldás. Létezik csak 2-től függő multiplikátor. tajarta a EE a :/—*.., E HET E E 2 alapján M(z) — [172/2, tehát [e[-42( 4 ) 4 $fe[-Azyy — 0 egzakt. A potenciól 2A l1 4 2te. oa f IA 09464 [c Az általános megoldás u(r.y) — C. amiből
5. Egyváltozós multiplikátorral tegyük egzakttá az y ln y + y sinh x + (x + yey)y′ = 0 diﬀeren- ciálegyenletet, majd oldjuk meg. Megoldás. Létezik csak y-tól függő multiplikátor: ∂(x+yey) Z −1 y dy = −ln y + C x+yey) ∂x −∂(y ln y+y sinh x) ∂y Z −1 ln \|M(y)\| = x − ∂y y ln y + y sinh x dy = alapján M(y) = 1/y, tehát ln y + sinh x + x y + ey y′ = 0 egzakt. A potenciál kereséséhez legyen a kezdőpont (0, 1), mert az origóban nem értelmes egyik komponens sem. Ekkor alapján M(y) = 1/y, tehát ln y + sinh x + x y + ey Z x Z y u(x, y) = Z 0 (ln 1 + sinh ξ) dξ + x η + eη dη = cosh x −1 + x ln y + ey −e. Az általános megoldás implicit alakban u(x, y(x)) = C, ebből y(x)-t nem lehet elemi függvénnyel kifejezni. További gyakorló feladatok 6. Határozzuk meg az y′ = sin(xy), y(0) = 0 kezdetiérték-probléma megoldásának kezdeti feltétel szerinti deriváltját. Megoldás. A konstans 0 függvény megoldja a kezdetiérték-problémát és a Picard-Lindelöftétel miatt ez az egyetlen megoldás. Legyen Y (x, 0, y0) az y(0) = y0 kezdeti feltételt kielégítő megoldás, D(x) ennek y0 szerinti deriváltja az y0 = 0 pontban. Ez a következő diﬀerenciálegyenletnek tesz eleget: D′(x) = ∂ ∂y0 ∂ ∂xY (x, 0, y0) ∂y0 sin(xY (x, 0, y0)) ∂y0 ∂x y0=0 ∂y0 y0=0 = cos(xY (x, 0, 0))xD(x) = xD(x). y0=0 Az kapott diﬀerenciálegyenlet szétválasztható: D′(x) D(x) = x, a két oldalt integrálva D(0) = 1 ﬁgyelembevételével ln D(x) = x2/2, vagyis D(x) = ex2/2. 7. Tekintsük az y′ 1 y′ 1 = xe−y1y2 y′ 2 = 1 −ey2 y′ 2 = 1 −ey2 diﬀerenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. Megoldás. Ha y1 = y2 = 0, akkor a jobb oldal 0 x értékétől függetlenül, tehát a konstans y1(x) = y2(x) = 0 megoldja a kezdetiérték-problémát. Legyen Y(x, 0, y0) az (y1(0), y2(0)) = y0 = (y0,1, y0,2) kezdeti feltételt kielégítő megoldás és Dij(x) az i. komponens y0,j szerinti deriváltja. Ez kielégíti a D′ ij(x) = ∂ ∂y0,j ∂ ∂xYi(x, 0, y0) y0=0 ∂y0,j fi(x, 0, Yi(x, 0, y0)) y0=0	5. Egyváltozós multiplikátorral tegyük egzakttá az y ln y + y sinh x + (x + ye[y])y[′] = 0 differenciálegyenletet, majd oldjuk meg. _Megoldás. Létezik csak y-tól függő multiplikátor:_ ln \|M (y)\| = � _∂(x+∂xye[y])_ _−_ _[∂][(][y][ ln][ y][+]∂y[y][ sinh][ x][)]_ dy = � _−1_ _y ln y + y sinh x_ _y_ [d][y][ =][ −] [ln][ y][ +][ C] alapján M (y) = 1/y, tehát ln y + sinh x + � _x_ _y[′]_ = 0 egzakt. A potenciál kereséséhez _y_ [+][ e][y][�] legyen a kezdőpont (0, 1), mert az origóban nem értelmes egyik komponens sem. Ekkor � _x_ � _y_ �x � _u(x, y) =_ dη 0 [(ln 1 + sinh][ ξ][) d][ξ][ +] 1 _η_ [+][ e][η] = cosh x − 1 + x ln y + e[y] _−_ _e._ Az általános megoldás implicit alakban u(x, y(x)) = C, ebből y(x)-t nem lehet elemi függvénnyel kifejezni. ## További gyakorló feladatok 6. Határozzuk meg az y[′] = sin(xy), y(0) = 0 kezdetiérték-probléma megoldásának kezdeti feltétel szerinti deriváltját. _Megoldás. A konstans 0 függvény megoldja a kezdetiérték-problémát és a Picard-Lindelöf-_ tétel miatt ez az egyetlen megoldás. Legyen Y (x, 0, y0) az y(0) = y0 kezdeti feltételt kielégítő megoldás, D(x) ennek y0 szerinti deriváltja az y0 = 0 pontban. Ez a következő differenciálegyenletnek tesz eleget: _∂_ _∂_ _∂_ _D[′](x) =_ = sin(xY (x, 0, y0)) _∂y0_ _∂x_ _[Y][ (][x,][ 0][, y][0][)]��y0=0_ _∂y0_ ��y0=0 = cos(xY (x, 0, 0))xD(x) = xD(x). Az kapott differenciálegyenlet szétválasztható: _D[′](x)_ _D(x) [=][ x,]_ a két oldalt integrálva D(0) = 1 figyelembevételével ln D(x) = x[2]/2, vagyis _D(x) = e[x][2][/][2]._ 7. Tekintsük az _y1[′]_ [=][ xe][−][y][1][y][2] _y2[′]_ [= 1][ −] _[e][y][2]_ differenciálegyenlet-rendszer y1(0) = y2(0) = 0 kezdeti feltételt kielégítő megoldását. Határozzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban. _Megoldás. Ha y1 = y2 = 0, akkor a jobb oldal 0 x értékétől függetlenül, tehát a konstans_ _y1(x) = y2(x) = 0 megoldja a kezdetiérték-problémát. Legyen Y(x, 0, y0) az (y1(0), y2(0)) =_ y0 = (y0,1, y0,2) kezdeti feltételt kielégítő megoldás és Dij(x) az i. komponens y0,j szerinti deriváltja. Ez kielégíti a _∂_ _Dij[′]_ [(][x][) =] _∂y0,j_ _∂_ _∂_ = _fi(x, 0, Yi(x, 0, y0))_ _∂x_ _[Y][i][(][x,][ 0][,][ y][0][)]��y0=0_ _∂y0,j_ ��y0=0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Egyváltozós multiplikátorral tegyük egzakttá az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ye</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀeren-</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciálegyenletet, majd oldjuk meg.</span></p> <p style="top:91.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Létezik csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-tól függő multiplikátor:</span></p> <p style="top:126.3pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:113.9pt;left:168.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:113.9pt;left:182.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ye</span></i><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">y</span></i></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:123.6pt;left:195.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂x</span></i></p> <p style="top:116.3pt;left:222.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> sinh</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:123.6pt;left:262.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">∂y</span></i></p> <p style="top:134.5pt;left:202.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:126.3pt;left:304.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:113.9pt;left:332.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:134.5pt;left:350.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup></p> <p style="top:159.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">/y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:149.3pt;left:295.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:157.8pt;left:302.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:166.6pt;left:302.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></sup></p> <p style="top:159.3pt;left:341.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egzakt. A potenciál kereséséhez</span></p> <p style="top:175.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">legyen a kezdőpont</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, mert az origóban nem értelmes egyik komponens sem. Ekkor</span></p> <p style="top:206.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:194.2pt;left:158.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:218.4pt;left:163.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln 1 + sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:194.2pt;left:271.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:218.4pt;left:277.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:190.7pt;left:288.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:214.8pt;left:297.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">η</span></i></sup></p> <p style="top:190.7pt;left:330.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:206.6pt;left:340.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></p> <p style="top:232.7pt;left:145.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e.</span></i></p> <p style="top:256.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az általános megoldás implicit alakban</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-t nem lehet elemi függ-</span></p> <p style="top:270.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vénnyel kifejezni.</span></p> <p style="top:303.3pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:327.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdetiérték-probléma megoldásának kezdeti</span></p> <p style="top:341.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">feltétel szerinti deriváltját.</span></p> <p style="top:360.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A konstans</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény megoldja a kezdetiérték-problémát és a Picard-Lindelöf-</span></p> <p style="top:374.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tétel miatt ez az egyetlen megoldás.</span></p> <p style="top:374.6pt;left:281.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt</span></p> <p style="top:389.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kielégítő megoldás,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ennek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szerinti deriváltja az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban. Ez a következő</span></p> <p style="top:403.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletnek tesz eleget:</span></p> <p style="top:434.3pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:426.2pt;left:160.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:442.5pt;left:155.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:426.2pt;left:178.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:442.5pt;left:174.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:417.9pt;left:241.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:449.9pt;left:244.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:434.3pt;left:267.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:426.2pt;left:287.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:442.5pt;left:282.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:434.3pt;left:303.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></p> <p style="top:417.9pt;left:385.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:449.9pt;left:388.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:462.4pt;left:140.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0))</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xD</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xD</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:485.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az kapott diﬀerenciálegyenlet szétválasztható:</span></p> <p style="top:507.2pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:523.5pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x,</span></i></sup></p> <p style="top:546.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a két oldalt integrálva</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ﬁgyelembevételével</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagyis</span></p> <p style="top:571.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:595.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Tekintsük az</span></p> <p style="top:619.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:625.5pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:636.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:642.9pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:660.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását. Ha-</span></p> <p style="top:675.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tározzuk meg a megoldás deriváltját a kezdeti feltétel szerint ebben a pontban.</span></p> <p style="top:693.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor a jobb oldal</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékétől függetlenül, tehát a konstans</span></p> <p style="top:708.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldja a kezdetiérték-problémát. Legyen</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)) =</span></p> <p style="top:722.4pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldás és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. komponens</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,j</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szerinti</span></p> <p style="top:736.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">deriváltja. Ez kielégíti a</span></p> <p style="top:767.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:773.8pt;left:116.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup></p> <p style="top:759.5pt;left:165.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:775.8pt;left:157.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,j</span></i></p> <p style="top:759.5pt;left:186.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:775.8pt;left:183.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:751.3pt;left:252.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:783.2pt;left:255.3pt;line-height:8.0pt"><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">y</span></b><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:767.6pt;left:279.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:759.5pt;left:302.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:775.8pt;left:294.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,j</span></i></p> <p style="top:767.6pt;left:318.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, Y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></p> <p style="top:751.3pt;left:414.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:783.2pt;left:417.5pt;line-height:8.0pt"><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">y</span></b><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_279.png	5. Egyváltozós multiplikátorral tegyűk egzakttá az ylny 4-ysinh r 4- (r 4 yey — 0 dítlere: ciálegyenletet, majd oldjuk meg. .Megoldás. Létezik csak y-tól fűggő multiplikátor: a. a skhi frzrernz kér alapján M(g) — 1/9. tehát Iny sh ( 4-ejyi — 0 eszakt. A pote meoshr-l4zinyie hyiC iál kereséséhez általános megoldás implicit alakban u(z, yíz)) ú függ- nyel kifejezni. C. ebből y(r)-t nem lehet e "További gyakorló feladatok 6. Határozzuk meg az 4 — sin(zy), y(0) — 0 kezdetiérték-problóma megoldásának kezdeti feltét A szerint deríváltját. att ez az egyetlen megoldás. Legyen Y(r.0.y) az y(0) — y9 kezdeti feltételt 0 0 ÖÖr 2 eos(zY(z.0.0YeDíz) — 2D). Az kapott differenciálegyenlet szétválasztható: Díz Díz) a két oldalt integrálya D(0) — 1 figyelembevételével ln Dír) — 12/2. vagyis De 7. Tekintsük az differenciálegyer tározzuk meg a megoldás deriváltját a kezdeti felt dszer y.(0) — y2(0) — 0 kezdeti feltételt kielégítő megoldását. Ha- el szerint ebben a pontban. .Megoldás, Ha y, — ya — 0, akkor a jobb oldal 0 2 értékétől fűggetlenül, tehát a konstans mmla) — wala) — 0 megoldja a kezdetiérték-problémát. Legyen Y (x. 0, yo) az (y1(0), v2(0)) 0 0 BDE - ömyöz XE - ph Kdye)
egyenletrendszert, ahol f1(x, y) = xe−y1y2 és f2(x, y) = 1 −ey2. A láncszabály alapján kifejtve az egyenletrendszer D′ 1 D′ 11(x) = xD21(x) D′ 12(x) = xD22(x) D′ 12(x) = xD22(x) D′ 21(x) = −D21(x) D′ 21(x) = −D21(x) D′ 22(x) = −D22(x) ′ 22(x) = −D22(x). A kezdeti feltétel Dij(x) = δij (Kronecker-delta, azaz D(0) = I az egységmátrix), ezzel az utolsó két egyenlet megoldható (mindkettő szétválasztható): D21(x) = 0 és D22(x) = e−x. Az első egyenlet ennek alapján D′ 11(x) = 0, tehát D11(x) = 1 konstans. Végül a e−x. Az első egyenlet ennek alapján D′ 11(x) = 0, tehát D11(x) = 1 konstans. Végül a második egyenlet D′ 12(x) = xe−x, amiből integrálással D12(x) = 1 −e−x −xe−x. Tehát a gy pj 11( ) , 11( ) g második egyenlet D′ 12(x) = xe−x, amiből integrálással D12(x) = 1 −e−x −xe−x. Tehát a deriváltmátrix D(x) = " # D11(x) D12(x) D21(x) D22(x) # 1 −e−x −xe−x . e−x 8. Határozzuk meg az y cosh x + (sinh x −2y)y′ = 0 diﬀerenciálegyenlet y(0) = 1 kezdeti feltételnek eleget tevő megoldását. Megoldás. Az egyenlet egzakt, mivel ∂(y cosh x) cosh x) ∂y = cosh x = ∂(sinh x −2y) ∂x ∂x Egy potenciál Z x 0 0 · cosh ξ dξ + Z y 0 (sinh x −2η) dη = y sinh x −y2. u(x, y) = Z y A kezdeti feltételt behelyettesítve u(x0, y0) = u(0, 1) = −1, tehát y(x) = 1 2 sinh x + q 4 + sinh2 x (A másodfokú egyenlet megoldásából . . .±√. . . adódik, a megfelelő előjelet a kezdeti feltétel határozza meg.) 9. Oldjuk meg az y cosh x+(sinh x−2y)y′ = 0 diﬀerenciálegyenletet y(0) = −1 kezdeti feltétel mellett. Megoldás. A diﬀerenciálegyenlet egzakt, mivel ∂ ∂yy cosh x = cosh x = ∂ ∂x sinh x A megoldás u(x, y(x)) = C alakban írható, ahol grad u(x, y) = y cos xi + (sinh x −2y)j és u(x0, y(x0)) = u(0, −1) = C. Egy lehetséges választás u(x, y) = y sinh x −y2, amiből C = −1 és így y(x) = 1 2 sinh x −1 2 q 4 + sinh2 x. (A másodfokú egyenlet megoldásából . . .±√. . . adódik, a megfelelő előjelet a kezdeti feltétel határozza meg.) 10. Határozzuk meg az x+y y + 2x+3y2 2y y′ = 0 diﬀerenciálegyenlet általános megoldását.	egyenletrendszert, ahol f1(x, y) = xe[−][y][1]y2 és f2(x, y) = 1 − _e[y][2]. A láncszabály alapján_ kifejtve az egyenletrendszer _D11[′]_ [(][x][) =][ xD][21][(][x][)] _D12[′]_ [(][x][) =][ xD][22][(][x][)] _D21[′]_ [(][x][) =][ −][D][21][(][x][)] _D22[′]_ [(][x][) =][ −][D][22][(][x][)][.] A kezdeti feltétel Dij(x) = δij (Kronecker-delta, azaz D(0) = I az egységmátrix), ezzel az utolsó két egyenlet megoldható (mindkettő szétválasztható): D21(x) = 0 és D22(x) = _e[−][x]. Az első egyenlet ennek alapján D11[′]_ [(][x][) = 0, tehát][ D][11][(][x][) = 1 konstans. Végül a] második egyenlet D12[′] [(][x][) =][ xe][−][x][, amiből integrálással][ D][12][(][x][) = 1][ −] _[e][−][x][ −]_ _[xe][−][x][. Tehát a]_ deriváltmátrix �1 1 − _e[−][x]_ _−_ _xe[−][x]_ 0 _e[−][x]_ � _._ _D(x) =_ �D11(x) _D12(x)�_ = _D21(x)_ _D22(x)_ 8. Határozzuk meg az y cosh x + (sinh x − 2y)y[′] = 0 differenciálegyenlet y(0) = 1 kezdeti feltételnek eleget tevő megoldását. _Megoldás. Az egyenlet egzakt, mivel_ _∂(y cosh x)_ = cosh x = _[∂][(sinh][ x][ −]_ [2][y][)]. _∂y_ _∂x_ Egy potenciál � _x_ � _y_ _u(x, y) =_ 0 [0][ ·][ cosh][ ξ][ d][ξ][ +] 0 [(sinh][ x][ −] [2][η][) d][η][ =][ y][ sinh][ x][ −] _[y][2][.]_ A kezdeti feltételt behelyettesítve u(x0, y0) = u(0, 1) = −1, tehát _y(x) = [1]_ 2 � � � sinh x + 4 + sinh[2] _x_ _._ (A másodfokú egyenlet megoldásából . . .±[√]. . . adódik, a megfelelő előjelet a kezdeti feltétel határozza meg.) 9. Oldjuk meg az y cosh x+(sinh x−2y)y[′] = 0 differenciálegyenletet y(0) = −1 kezdeti feltétel mellett. _Megoldás. A differenciálegyenlet egzakt, mivel_ _∂_ _∂y_ _[y][ cosh][ x][ = cosh][ x][ =][ ∂]∂x_ [sinh][ x] A megoldás u(x, y(x)) = C alakban írható, ahol grad u(x, y) = y cos xi + (sinh x − 2y)j és u(x0, y(x0)) = u(0, −1) = C. Egy lehetséges választás u(x, y) = y sinh x − _y[2], amiből_ _C = −1 és így_ _y(x) = [1]_ 2 [sinh][ x][ −] [1]2 � 4 + sinh[2] _x._ (A másodfokú egyenlet megoldásából . . .±[√]. . . adódik, a megfelelő előjelet a kezdeti feltétel határozza meg.) 10. Határozzuk meg az _[x][+][y]_ + [2][x][+3][y][2] _y[′]_ = 0 differenciálegyenlet általános megoldását. _y_ 2y -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszert, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A láncszabály alapján</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kifejtve az egyenletrendszer</span></p> <p style="top:99.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:105.9pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xD</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:117.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:123.3pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xD</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:134.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:140.8pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:152.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:158.2pt;left:116.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:178.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> δ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ij</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (Kronecker-delta, azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az egységmátrix), ezzel</span></p> <p style="top:192.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az utolsó két egyenlet megoldható (mindkettő szétválasztható):</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:207.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az első egyenlet ennek alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:213.3pt;left:284.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> konstans. Végül a</span></sup></p> <p style="top:221.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">második egyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:227.7pt;left:179.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből integrálással</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Tehát a</span></sup></p> <p style="top:236.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">deriváltmátrix</span></p> <p style="top:266.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:250.8pt;left:148.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:259.4pt;left:153.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">11</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:259.4pt;left:198.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:273.9pt;left:153.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">21</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:273.9pt;left:198.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">22</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:250.8pt;left:232.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:266.7pt;left:241.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:250.8pt;left:254.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:259.4pt;left:260.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:259.4pt;left:275.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:273.9pt;left:260.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:273.9pt;left:305.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:250.8pt;left:352.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:266.7pt;left:359.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:300.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti</span></p> <p style="top:314.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">feltételnek eleget tevő megoldását.</span></p> <p style="top:334.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet egzakt, mivel</span></p> <p style="top:358.3pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:374.6pt;left:128.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i></p> <p style="top:366.4pt;left:167.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:374.6pt;left:256.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i></p> <p style="top:366.4pt;left:300.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:397.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Egy potenciál</span></p> <p style="top:427.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:414.8pt;left:156.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:439.1pt;left:161.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:414.8pt;left:246.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i></p> <p style="top:439.1pt;left:251.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">η</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:457.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltételt behelyettesítve</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:489.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:497.2pt;left:145.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:476.0pt;left:154.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:489.0pt;left:161.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:476.2pt;left:205.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:489.0pt;left:215.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 + sinh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:476.0pt;left:270.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:489.0pt;left:279.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:519.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(A másodfokú egyenlet megoldásából</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> . . .</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">. . .</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adódik, a megfelelő előjelet a kezdeti feltétel</span></p> <p style="top:534.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">határozza meg.)</span></p> <p style="top:553.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+(sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:567.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:587.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A diﬀerenciálegyenlet egzakt, mivel</span></p> <p style="top:610.6pt;left:110.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂</span></i></p> <p style="top:626.9pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂y</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ∂</span></i></sup></p> <p style="top:626.9pt;left:224.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">∂x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:651.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban írható, ahol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> grad</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">i</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">j</span></b></p> <p style="top:665.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Egy lehetséges választás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x, y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span></p> <p style="top:679.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és így</span></p> <p style="top:710.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:718.8pt;left:145.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:718.8pt;left:199.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:697.8pt;left:206.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:710.6pt;left:216.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 + sinh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></p> <p style="top:740.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(A másodfokú egyenlet megoldásából</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> . . .</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">. . .</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adódik, a megfelelő előjelet a kezdeti feltétel</span></p> <p style="top:754.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">határozza meg.)</span></p> <p style="top:775.6pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Határozzuk meg az</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup></p> <p style="top:782.9pt;left:186.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:775.6pt;left:200.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:782.9pt;left:223.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:775.6pt;left:242.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_280.png	egyenletrendszert, ahol fi(z.y) ya és felnny) — 1— é". A láncszabály alapján. D4(e) — eDalz) Djale) — xDodlr) Dule) - —Dalr) Diale) — —D. A kezdeti feltétel Du(z) — ő; (Kronecker-delta, azaz D(0) — I az egységmátris), ezzel az utolsó két egyenlet megoldható (mindkettő szétválasztható): Dyfz) — 0 és Des(z) €77. Az első egyenlet ennek alapján Dí,(z) — 0. tehát Du(r) — 1 konstans. Végül a második egyenlet Día(2) — ze-2, amiből intográlással Da(2) — 1— €-7 —2ze-t. Tehát a 99- [9 pok ] 8. Határozzuk meg az yeoshz 4. (sinh: — 2yy — 0 dífferenciól kezdeti feltételnek eleget tevő megoldását. egyen .Megoldás. Az egyenlet egzakt, mivel ölveohz) , a - Tinhz — 29) 0 E Egy potenciál üley) — [/0-coshgds 4. [/tinh — 2n)dr — ysinhz — v A kezdeti felt 1, tehát telt behelyettesítve u(zo, ) — 10. 1 oa v9-3(8be s Vs (A másodfokú egyen 4 a kezdeti feltétel határozza meg.) et megoldlásából . .. /— adódik, a megfelelő előjel 9. Oldjuk meg az ycosh 2- (sinh — 2)Y mellett. diflerenciálogyenletet y(0) — —1 kezdeti feltétel .Megoldás. A difforenciálegyenlet egzakt, mivel. : ! A megoldás ulz, v(e)) — C alakban írható, ahol gradul, 9) — ycoszi 4 (sinhe — 2) C ésígy 1 1T wl) — 25lnhe — V sitéz et megoldásából . .. £ /— adódik, a megfelelő elő 10. Határozzuk meg az " 4 — 0 díferenciálegyenlet általános megoldását. 4
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 9. feladatsor: Állandók variálása, megoldás sorfejtéssel 1. Határozzuk meg az y′ + 2xy = 2xe−x2 diﬀerenciálegyenlet általános megoldását. 2. Határozzuk meg az " −1 2e−2x −e2x 4 y′ = y + " −6e−2x# diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (ex, e3x) és y2(x) = (2, e2x) megoldásai. 3. Határozzuk meg az y′ = 1+x2 0 1 1 1+x2 y + # 1+x2 diﬀerenciálegyenlet-rendszer általános megoldását. 4. Határozzuk meg az xy′′ −(x + 1)y′ + y = x2ex diﬀerenciálegyenlet általános megoldását, ha tudjuk, hogy y1(x) = ex és y2(x) = x + 1 megoldja a hozzá tartozó homogén egyenletet. 5. Oldjuk meg sorfejtéssel az y′ = x+y diﬀerenciálegyenletet y(0) = 0 kezdeti feltétel mellett. diﬀerenciálegyenlet-rendszer általános megoldását. 4. Határozzuk meg az xy′′ −(x + 1)y′ + y = x2ex diﬀerenciálegyenlet általános megoldását, ha További gyakorló feladatok 6. Határozzuk meg az y′ −(tan x + ctg x)y = −4 sin2 x diﬀerenciálegyenlet általános megoldá- sát. 7. Oldjuk meg az xy′ −y = x3 + 1 diﬀerenciálegyenletet y(2) = 5 kezdeti feltétel mellett. 8. Határozzuk meg az y′ + y = e−x diﬀerenciálegyenlet általános megoldását. 9. Oldjuk meg az y′ + y cos x = sin x cos x diﬀerenciálegyenletet y(0) = 1 kezdeti feltétel mellett. 10. Határozzuk meg az y′ = 1 −x2 −x " # y + x 1 diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (1 + x2, x) és y2(x) = (x, 1) megoldásai. 11. Oldjuk meg az y′ = 1+x2 x √ 1+x2 1+x2 x2 √ 1+x2 1+x2 y + diﬀerenciálegyenlet-rendszert y(0) = (0, 1) kezdeti feltétel mellett. 12. Határozzuk meg az y′′ −y = 2 x −2x log x diﬀerenciálegyenlet általá 12. Határozzuk meg az y′′ −y = 2 x −2x log x diﬀerenciálegyenlet általános megoldását (x > 0), ha tudjuk, hogy y1(x) = ex és y2(x) = e−x megoldja a hozzá tartozó homogén egyenletet. 13. Oldjuk meg az xy′′′ + 2y′′ = 1 x diﬀerenciálegyenletet y(1) = 1, y′(1) = y′′(1) = 0 kezdeti 13. Oldjuk meg az xy′′′ + 2y′′ = 1 x diﬀerenciálegyenletet y(1) = 1, y′(1) = y′′(1) = 0 kezdeti feltétellel. 14. Sorfejtés segítségével határozzuk meg az (1 −x)y′′ + xy′ −y = 0 diﬀerenciálegyenletet y(0) = y′(0) = 1 feltételt kielégítő megoldását. 15. Sorfejtés segítségével határozzuk meg az (1 −2x)y′′ + (8x3 + 4x)y′ + (−12x2 + 4x −2)y = 0 diﬀerenciálegyenlet y(0) = 1, y′(0) = 0 kezdeti feltételt kielégítő megoldását.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 9. feladatsor: Állandók variálása, megoldás sorfejtéssel 1. Határozzuk meg az y[′] + 2xy = 2xe[−][x][2] differenciálegyenlet általános megoldását. 2. Határozzuk meg az � y + �−6e[−][2][x] 0 � y[′] = � _−1_ 2e[−][2][x] _−e[2][x]_ 4 differenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (e[x], e[3][x]) és y2(x) = (2, e[2][x]) megoldásai. 3. Határozzuk meg az � 1 1+x[2] 1 y[′] = � 1 0 1+x[2] 1 1 1+x[2] � y + � differenciálegyenlet-rendszer általános megoldását. 4. Határozzuk meg az xy[′′] _−_ (x + 1)y[′] + y = x[2]e[x] differenciálegyenlet általános megoldását, ha tudjuk, hogy y1(x) = e[x] és y2(x) = x + 1 megoldja a hozzá tartozó homogén egyenletet. 5. Oldjuk meg sorfejtéssel az y[′] = x + _y differenciálegyenletet y(0) = 0 kezdeti feltétel mellett._ ## További gyakorló feladatok 6. Határozzuk meg az y[′] _−_ (tan x + ctg x)y = −4 sin[2] _x differenciálegyenlet általános megoldá-_ sát. 7. Oldjuk meg az xy[′] _−_ _y = x[3]_ + 1 differenciálegyenletet y(2) = 5 kezdeti feltétel mellett. 8. Határozzuk meg az y[′] + y = e[−][x] differenciálegyenlet általános megoldását. 9. Oldjuk meg az y[′] + y cos x = sin x cos x differenciálegyenletet y(0) = 1 kezdeti feltétel mellett. 10. Határozzuk meg az �x 1 � � y + y[′] = �x 1 − _x[2]_ 1 _−x_ differenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (1 + x[2], x) és y2(x) = (x, 1) megoldásai. 11. Oldjuk meg az � 1 1+x[2] _−_ _√_ _x[2]_ 1+x[2] � y[′] = � _x_ 0 1+x[2] _√_ _x_ _x_ 1+x[2] 1+x[2] � y + differenciálegyenlet-rendszert y(0) = (0, 1) kezdeti feltétel mellett. 12. Határozzuk meg az y[′′] _−_ _y =_ [2] _x_ _[−]_ [2][x][ log][ x][ differenciálegyenlet általános megoldását (][x >][ 0),] ha tudjuk, hogy y1(x) = e[x] és y2(x) = e[−][x] megoldja a hozzá tartozó homogén egyenletet. 13. Oldjuk meg az xy[′′′] + 2y[′′] = [1] _x_ [differenciálegyenletet][ y][(1) = 1,][ y][′][(1) =][ y][′′][(1) = 0 kezdeti] feltétellel. 14. Sorfejtés segítségével határozzuk meg az (1 − _x)y[′′]_ + xy[′] _−_ _y = 0 differenciálegyenletet_ _y(0) = y[′](0) = 1 feltételt kielégítő megoldását._ 15. Sorfejtés segítségével határozzuk meg az (1 − 2x)y[′′] + (8x[3] + 4x)y[′] + (−12x[2] + 4x − 2)y = 0 differenciálegyenlet y(0) = 1, y[′](0) = 0 kezdeti feltételt kielégítő megoldását. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:94.0pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">9. feladatsor: Állandók variálása, megoldás sorfejtéssel</span></b></p> <p style="top:130.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:147.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg az</span></p> <p style="top:175.8pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:159.9pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:168.5pt;left:142.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:168.5pt;left:172.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:183.0pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:183.0pt;left:182.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:159.9pt;left:199.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:175.8pt;left:207.3pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:159.9pt;left:228.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:168.5pt;left:234.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:183.0pt;left:250.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:159.9pt;left:271.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:204.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek</span></p> <p style="top:218.8pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldásai.</span></p> <p style="top:235.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Határozzuk meg az</span></p> <p style="top:263.9pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:248.0pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:254.4pt;left:146.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:263.3pt;left:139.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:256.0pt;left:178.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:271.0pt;left:146.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:269.5pt;left:179.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:278.3pt;left:171.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:248.0pt;left:192.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:263.9pt;left:200.0pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:248.0pt;left:221.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:254.7pt;left:236.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:263.6pt;left:228.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:271.3pt;left:235.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:248.0pt;left:249.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:292.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását.</span></p> <p style="top:308.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet általános megoldását, ha</span></p> <p style="top:323.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tudjuk, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldja a hozzá tartozó homogén egyenletet.</span></p> <p style="top:339.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Oldjuk meg sorfejtéssel az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:371.9pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:396.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ctg</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldá-</span></p> <p style="top:410.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sát.</span></p> <p style="top:426.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2) = 5</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:443.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:459.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:474.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:490.6pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Határozzuk meg az</span></p> <p style="top:519.2pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:503.2pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:511.9pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:511.9pt;left:154.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:526.3pt;left:138.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:526.3pt;left:162.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:503.2pt;left:186.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:519.2pt;left:194.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:503.2pt;left:215.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:511.9pt;left:221.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:526.3pt;left:222.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:503.2pt;left:228.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:547.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek</span></p> <p style="top:562.1pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldásai.</span></p> <p style="top:578.5pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Oldjuk meg az</span></p> <p style="top:608.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:592.4pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:598.4pt;left:150.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:607.2pt;left:142.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:599.9pt;left:185.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:613.5pt;left:150.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:616.5pt;left:139.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:623.2pt;left:146.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:613.5pt;left:185.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:622.3pt;left:178.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:592.4pt;left:199.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:608.3pt;left:207.0pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:592.4pt;left:228.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:597.8pt;left:251.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:606.6pt;left:243.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:615.6pt;left:234.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:614.1pt;left:253.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:617.1pt;left:244.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:623.8pt;left:251.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:592.4pt;left:272.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:638.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:655.3pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:662.6pt;left:226.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> log</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x ></span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">),</span></sup></p> <p style="top:669.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha tudjuk, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldja a hozzá tartozó homogén egyenletet.</span></p> <p style="top:686.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:693.4pt;left:230.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti</span></sup></p> <p style="top:700.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">feltétellel.</span></p> <p style="top:717.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">14. Sorfejtés segítségével határozzuk meg az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span></p> <p style="top:731.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> feltételt kielégítő megoldását.</span></p> <p style="top:747.9pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15. Sorfejtés segítségével határozzuk meg az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:762.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> </div>	page_281.png	Matematika A3 gyakorlat Energetika és Mechatronika HSc szakok, 2016/17 ősz 9. feladatsor: Állandók variálása, megoldás sorfejtéssel 1. Határozzuk meg az y/ -- 2zy — 2ze-? dífferenciálegyenlet általános megoldását. 2. Határozzuk meg az [ 2e"y a [6] velle 4 14 differenciálegyer dszer általános megoldását, ha a hozzá tartozó homogén rendszernek l — ( e) és yalr) — (2.e22) megoldásai. 3. Határozzuk meg az differenciálegyer dszer általános megoldását. 5. Oldjuk meg sorfejtéssel az y — x 4. y differenciálegyenletet 9(0) — 0 kezdeti feltétel mellett. "További gyakorló feladatok 6. Határozzuk meg az / — (tanz 4-etgz)y — —dsinő r díífes sát álogyenlet általános megoldá- 7. Oldjuk meg az 2y/ — y — 29 41 dítfer letet y(2) — 5 kezdeti feltétel mellett 9. Oldjuk meg az / 4 yeosz — sinzeosz diíferenciálegyenletet y(0) — 1 kezdeti feltétel mellett. iálogye 10. Határozzuk meg az MRTE ] 11. Oldjuk meg az 12. Határozzuk meg az y" — y — 2 mciálegyenlet általános megoldását (r — 0). ha tudjuk, hogy m(z) — € és vlz) — €-? megoldja a hozzá tartozó homogén egyenletet V) — /0) — 0 kezdeti dszer általános megoldását, ha a hozzá tartozó homogén rendszernek Vale) — (z.1) megoldásai. differenciálegyet 2xlog dítfet 13. Oldjuk meg az 247 4. 24" — 1 differenciálegyenletet y(1) feltétellel. 14. Sorfejtés segítségével határozzuk meg az (1— 2)y" 4 24/— y w(0) — w(0) — 1 feltételt kielésítő megoldását. 15. Sorfejtés segítségével határozzuk meg az (1— 2799" 4. (82? 4-4rjy4(-12724-4r—2jy— 0 dilferenciálegyenlet y(0) — 1,9y(0) — 0 kezdeti feltételt kielégítő megoldását 0 differenciálogyenletet
A kezdeti feltételből 1 = y(0) = a0 és 1 = y′(0) = a1, az sorfejtés első tagjából 0 = 2a2 −a0, azaz a2 = 1 2, a többi tagból pedig k ≥1 esetén a ak+2 = (k + 1)kak+1 −(k −1)ak (k + 1)(k + 2) rekurziót kapjuk. Ebből az első néhány tagot meghatározva sejthetjük, hogy ak = 1 k!, behelyettesítve ellenőrizzük: k! (k + 1)k 1 (k+1)! −(k −1) 1 ak+2 = 1)k 1 (k+1)! −(k −1) 1 k! (k + 1)(k + 2) = (k −(k −1)) 1 k! (k + 1)(k + 2) (k (k 1))k! (k + 1)(k + 2) = 1 (k + 2)! A kapott hatványsor minden x-re abszolút konvergens, összege y(x) = ex, ami megoldja a kezdetiérték-problémát. 15. Sorfejtés segítségével határozzuk meg az y′′ −xy′ + 4y = 0 diﬀerenciálegyenlet y(0) = 3, y′(0) = 0 kezdeti feltételt kielégítő megoldását. Megoldás. A megoldást X akxk k=0 y(x) = alakban keressük, ennek deriváltjai y′(x) = y′′(x) = X kakxk−1 k=1 ∞ X k(k −1)akxk−2. k=2 Helyettesítsük be az egyenletbe: 0 = k=2 k(k −1)akxk−2 −x k=1 kakxk−1 + 4 X akxk k=0 ∞ X (k + 2)(k + 1)ak+2xk − k=0 k=1 kakxk + 4 X akxk k=0 = 2a2 + 4a0 + (k + 2)(k + 1)ak+2 −kak + 4ak xk. k=1 A jobb oldalon álló hatványsor akkor azonosan 0, ha minden együttható 0, ebből a következő rekurziót kapjuk: ak+2 = k−4 (k+2)(k+1)ak ha k ≥1 −2a0 ha k = 0. A kezdeti feltétel alapján a0 = y(0) = 3 és a1 = y′(0) = 0, innen a rekurzió alapján meghatározható a többi együttható. a1 = 0 miatt az összes páratlan sorszámú is 0 lesz. Másrészt k = 4 esetben a rekurzióból a6 = 0 következik, és emiatt ak = 0 minden k ≥6 páros számra is. Következésképp a hatványsor valójában egy legfejlebb negyedfokú polinom. Az együtthatók a2 = −2·3 = −6 és a4 = −2 4·3 ·(−6) = 1. A kezdetiérték-probléma megoldása y(x) = x4 −6x2 + 3. A kezdeti feltétel alapján a0 = y(0) = 3 és a1 = y′(0) = 0, innen a rekurzió alapján meghatározható a többi együttható. a1 = 0 miatt az összes páratlan sorszámú is 0 lesz. Másrészt k = 4 esetben a rekurzióból a6 = 0 következik, és emiatt ak = 0 minden k ≥6 páros számra is. Következésképp a hatványsor valójában egy legfejlebb negyedfokú polinom. Az együtthatók a2 = −2·3 = −6 és a4 = −2 4·3 ·(−6) = 1. A kezdetiérték-probléma megoldása 4 2 Megjegyzés. Az y′′ −xy′ + λy = 0 (λ ∈R paraméter) diﬀerenciálegyenlet neve Hermitediﬀerenciálegyenlet. Ha λ ∈N, akkor létezik λ fokú polinom megoldás, egyébként pedig minden megoldás exponenciálisan nő.	A kezdeti feltételből 1 = y(0) = a0 és 1 = y[′](0) = a1, az sorfejtés első tagjából 0 = 2a2 − _a0,_ azaz a2 = [1]2 [, a többi tagból pedig][ k][ ≥] [1 esetén a] _ak+2 = [(][k][ + 1)][ka][k][+1][ −]_ [(][k][ −] [1)][a][k] (k + 1)(k + 2) rekurziót kapjuk. Ebből az első néhány tagot meghatározva sejthetjük, hogy ak = _k1!_ [,] behelyettesítve ellenőrizzük: _ak+2 =_ (k + 1)k (k+1)!1 _[−]_ [(][k][ −] [1)][ 1]k! = [(][k][ −] [(][k][ −] [1))][ 1]k! 1 (k + 1)(k + 2) (k + 1)(k + 2) [=] (k + 2)! A kapott hatványsor minden x-re abszolút konvergens, összege y(x) = e[x], ami megoldja a kezdetiérték-problémát. 15. Sorfejtés segítségével határozzuk meg az y[′′] _−_ _xy[′]_ + 4y = 0 differenciálegyenlet y(0) = 3, y[′](0) = 0 kezdeti feltételt kielégítő megoldását. _Megoldás. A megoldást_ _y(x) =_ _∞_ � _akx[k]_ _k=0_ alakban keressük, ennek deriváltjai _y[′](x) =_ _y[′′](x) =_ _∞_ � _kakx[k][−][1]_ _k=1_ _∞_ � _k(k −_ 1)akx[k][−][2]. _k=2_ Helyettesítsük be az egyenletbe: _∞_ _∞_ _∞_ 0 = � _k(k −_ 1)akx[k][−][2] _−_ _x_ � _kakx[k][−][1]_ + 4 � _akx[k]_ _k=2_ _k=1_ _k=0_ _∞_ _∞_ _∞_ = �(k + 2)(k + 1)ak+2x[k] _−_ � _kakx[k]_ + 4 � _akx[k]_ _k=0_ _k=1_ _k=0_ _∞_ = 2a2 + 4a0 + � �(k + 2)(k + 1)ak+2 − _kak + 4ak�x[k]._ _k=1_ A jobb oldalon álló hatványsor akkor azonosan 0, ha minden együttható 0, ebből a következő rekurziót kapjuk: _ak+2 =_   −2a0 ha k = 0. _k−4_ ha k ≥ 1 (k+2)(k+1) _[a][k]_ A kezdeti feltétel alapján a0 = y(0) = 3 és a1 = y[′](0) = 0, innen a rekurzió alapján meghatározható a többi együttható. a1 = 0 miatt az összes páratlan sorszámú is 0 lesz. Másrészt k = 4 esetben a rekurzióból a6 = 0 következik, és emiatt ak = 0 minden k ≥ 6 páros számra is. Következésképp a hatványsor valójában egy legfejlebb negyedfokú polinom. Az együtthatók a2 = −2·3 = −6 és a4 = _[−]4·[2]3_ _[·][(][−][6) = 1. A kezdetiérték-probléma megoldása]_ _y(x) = x[4]_ _−_ 6x[2] + 3. _Megjegyzés. Az y[′′]_ _−_ _xy[′]_ + λy = 0 (λ ∈ paraméter) differenciálegyenlet neve HermiteR differenciálegyenlet. Ha λ ∈, akkor létezik λ fokú polinom megoldás, egyébként pedig N minden megoldás exponenciálisan nő. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltételből</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az sorfejtés első tagjából</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:80.8pt;left:131.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a többi tagból pedig</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén a</span></sup></p> <p style="top:104.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:112.7pt;left:172.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)</span></p> <p style="top:135.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rekurziót kapjuk.</span></p> <p style="top:135.3pt;left:176.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből az első néhány tagot meghatározva sejthetjük, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:133.8pt;left:528.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:142.6pt;left:527.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">!</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:149.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">behelyettesítve ellenőrizzük:</span></p> <p style="top:182.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:171.7pt;left:145.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></p> <p style="top:170.1pt;left:199.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:179.0pt;left:189.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1)!</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:179.0pt;left:266.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">!</span></p> <p style="top:190.4pt;left:174.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)</span></p> <p style="top:182.2pt;left:279.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1))</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:181.0pt;left:360.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">!</span></p> <p style="top:190.4pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:174.1pt;left:403.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:190.4pt;left:386.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)!</span></p> <p style="top:212.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kapott hatványsor minden</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re abszolút konvergens, összege</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami megoldja a</span></p> <p style="top:226.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kezdetiérték-problémát.</span></p> <p style="top:244.6pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15. Sorfejtés segítségével határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></p> <p style="top:259.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását.</span></p> <p style="top:277.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megoldást</span></p> <p style="top:306.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:296.8pt;left:147.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:298.2pt;left:144.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:321.9pt;left:144.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:306.6pt;left:161.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:337.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük, ennek deriváltjai</span></p> <p style="top:367.2pt;left:108.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:357.5pt;left:152.7pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:358.9pt;left:149.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:382.6pt;left:149.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:367.2pt;left:166.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:401.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:392.0pt;left:152.7pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:393.4pt;left:149.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:417.1pt;left:149.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2</span></p> <p style="top:401.8pt;left:166.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:433.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Helyettesítsük be az egyenletbe:</span></p> <p style="top:462.4pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span></p> <p style="top:452.7pt;left:133.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:454.1pt;left:130.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:477.8pt;left:130.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2</span></p> <p style="top:462.4pt;left:147.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:452.7pt;left:250.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:454.1pt;left:247.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:477.8pt;left:247.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:462.4pt;left:264.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></p> <p style="top:452.7pt;left:330.7pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:454.1pt;left:327.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:477.8pt;left:327.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:462.4pt;left:344.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:497.0pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:487.2pt;left:133.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:488.6pt;left:130.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:512.3pt;left:130.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:497.0pt;left:145.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:487.2pt;left:269.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:488.6pt;left:266.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:512.3pt;left:265.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:497.0pt;left:283.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></p> <p style="top:487.2pt;left:338.4pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:488.6pt;left:335.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:512.3pt;left:334.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:497.0pt;left:352.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:531.5pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:521.7pt;left:195.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:523.1pt;left:192.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:546.9pt;left:192.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:521.5pt;left:209.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:531.5pt;left:215.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:521.5pt;left:373.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:531.5pt;left:379.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:562.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A jobb oldalon álló hatványsor akkor azonosan</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ha minden együttható</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ebből a következő</span></p> <p style="top:577.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rekurziót kapjuk:</span></p> <p style="top:610.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:590.8pt;left:144.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:599.8pt;left:144.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:617.7pt;left:144.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:600.0pt;left:168.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></p> <p style="top:608.8pt;left:154.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2)(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1)</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:601.5pt;left:222.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:619.8pt;left:153.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:619.8pt;left:222.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:643.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltétel alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, innen a rekurzió alapján</span></p> <p style="top:658.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">meghatározható a többi együttható.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> miatt az összes páratlan sorszámú is</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> lesz.</span></p> <p style="top:672.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Másrészt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetben a rekurzióból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> következik, és emiatt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> minden</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:687.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">páros számra is. Következésképp a hatványsor valójában egy legfejlebb negyedfokú polinom.</span></p> <p style="top:701.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az együtthatók</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:708.9pt;left:286.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">·</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6) = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A kezdetiérték-probléma megoldása</span></sup></p> <p style="top:716.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:734.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megjegyzés.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméter) diﬀerenciálegyenlet neve Hermite-</span></p> <p style="top:748.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet. Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">N</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor létezik</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> fokú polinom megoldás, egyébként pedig</span></p> <p style="top:763.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">minden megoldás exponenciálisan nő.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> </div>	page_282.png	15. A kezdeti feltételből 1 — y(0) — ag és 1— y(0) 1. a többi tagból pedig k 2 1 csetén a és első tagjából 0 — 2az — a. , az sorfej a [k Dkara — (E 1ox AÁRÁN CTSE TTCESEJT kapjuk. Ebből az első né behelyettesítve ellenőrizzűk: (köVkely-(k-0$ . ($-(k- 4 [EFTEE] (FZE 2) Ő TR A kapott hatványsor minden 2-re abszolút konvergens, összege y(z) — €", ami megoldja a kezdetiérték-problémát. Sorfejtés 3.(0) "Megoldás. A megoldást voa-azt segítségével határozzuk meg az y" — 2 4 4y — 0 differenciálegyenlet y(0) — kezdeti feltételt kielégítő megoldását vE Ék..u*' v kik(l. Va. Helyettesítsük be az egyenletbe: S(k YE Var - V kazzt 445 aszt 2a 4409 452 ((k 4241r — kar 4 da)et. A jobb oldalon álló hatványsor akkor azonosan 0, ha minden együttható 0, ebből a következő rekurziót kapjuk: aoz- (Őh hakz ] 2a) hakzó. A kezdeti feltétel alapján ap — y(0) — 3 és a — y(0) — 0. irnen a rekurzió alapján Másrészt k — 4 esetben a rekurzióból ag — 0 következik, és emiatt az — 0 minden £ 2 6 vWeyz t6 43. Megjegyzés. Az 4" — ay 4 Ay — 0 (X € IR paramétet) differenciálegyenlet neve Hermite-
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 9. feladatsor: Állandók variálása, megoldás sorfejtéssel (megoldás) 1. Határozzuk meg az y′ + 2xy = 2xe−x2 diﬀerenciálegyenlet általános megoldását. Megoldás. Az egyenlet elsőrendű lineáris, először a hozzá tartozó homogén egyenletet oldjuk meg, ami szétválasztható: y′ y = −2x. A két oldal integrálása után ln \|y(x)\| = −x2 + C, vagy átrendezve és ±eC helyett C-t írva y(x) = Ce−x2 adódik. Az inhomogén egyenlet megoldását az állandók variálásának módszere szerint y(x) = c(x)e−x2 alakban keressük. Az egyenlet bal oldalába helyettesítjük: y′(x) + 2xy(x) = c′(x)e−x2 −c(x)2xe−x2 + 2xc(x)e−x2 = c′(x)e−x2. A kapott kifejezésnek a jobb oldallal kell megegyeznie, tehát c′(x) = 2x, vagyis c(x) = x2+C, ahol C tetszőleges konstans. Az egyenlet általános megoldása tehát y(x) = x2e−x2 + Ce−x2. 2. Határozzuk meg az " −1 2e−2x −e2x 4 y′ = y + " −6e−2x# diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (ex, e3x) és y2(x) = (2, e2x) megoldásai. Megoldás. A két függvény lineárisan független, tehát a homogén egyenlet általános megoldása y(x) = C1y1(x) + C2y2(x). Legyen U(x) az a mátrix, aminek oszlopai az y1(x) és y2(x) vektorok, A(x) pedig az y előtt álló együtthatómátrix. Az állandók variálásának módszere szerint az inhomogén egyenlet megoldását y(x) = U(x)c(x) alakban keressük. U ′(x) = A(x)U(x) alapján a derivált y′(x) = U ′(x)c(x) + U(x)c′(x) = A(x)U(x)c(x) + U(x)c′(x) = A(x)y(x) + U(x)c′(x), tehát y(x) akkor oldja meg az egyenletet, ha c′(x) = U(x)−1 " −6e−2x# teljesül. Számítsuk ki U(x) inverzét: U(x)−1 = " #−1 ex 2 e3x e2x ex · e2x −2e3x " # e2x −2 −e3x ex " # −e−x 2e−3x , 1 −e−2x	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 9. feladatsor: Állandók variálása, megoldás sorfejtéssel (megoldás) 1. Határozzuk meg az y[′] + 2xy = 2xe[−][x][2] differenciálegyenlet általános megoldását. _Megoldás. Az egyenlet elsőrendű lineáris, először a hozzá tartozó homogén egyenletet oldjuk_ meg, ami szétválasztható: _y[′]_ _y_ [=][ −][2][x.] A két oldal integrálása után ln \|y(x)\| = −x[2] + C, vagy átrendezve és ±e[C] helyett C-t írva _y(x) = Ce[−][x][2]_ adódik. Az inhomogén egyenlet megoldását az állandók variálásának módszere szerint y(x) = _c(x)e[−][x][2]_ alakban keressük. Az egyenlet bal oldalába helyettesítjük: _y[′](x) + 2xy(x) = c[′](x)e[−][x][2]_ _−_ _c(x)2xe[−][x][2]_ + 2xc(x)e[−][x][2] = c[′](x)e[−][x][2]. A kapott kifejezésnek a jobb oldallal kell megegyeznie, tehát c[′](x) = 2x, vagyis c(x) = _x[2]_ +C, ahol C tetszőleges konstans. Az egyenlet általános megoldása tehát y(x) = x[2]e[−][x][2] + _Ce[−][x][2]._ 2. Határozzuk meg az �−6e[−][2][x] 0 y[′] = � _−1_ 2e[−][2][x] _−e[2][x]_ 4 � y + � differenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (e[x], e[3][x]) és y2(x) = (2, e[2][x]) megoldásai. _Megoldás. A két függvény lineárisan független, tehát a homogén egyenlet általános meg-_ oldása y(x) = C1y1(x) + C2y2(x). Legyen U (x) az a mátrix, aminek oszlopai az y1(x) és y2(x) vektorok, A(x) pedig az y előtt álló együtthatómátrix. Az állandók variálásának módszere szerint az inhomogén egyenlet megoldását y(x) = U (x)c(x) alakban keressük. _U_ _[′](x) = A(x)U_ (x) alapján a derivált y[′](x) = U _[′](x)c(x) + U_ (x)c[′](x) = A(x)U (x)c(x) + U (x)c[′](x) = A(x)y(x) + U (x)c[′](x), tehát y(x) akkor oldja meg az egyenletet, ha �−6e[−][2][x] c[′](x) = U (x)[−][1] 0 � teljesül. Számítsuk ki U (x) inverzét: �−e[−][x] 2e[−][3][x] 1 _−e[−][2][x]_ � _,_ �−1 1 = _e[x]_ _· e[2][x]_ _−_ 2e[3][x] � _e[2][x]_ _−2�_ = _−e[3][x]_ _e[x]_ _U_ (x)[−][1] = � _e[x]_ 2 _e[3][x]_ _e[2][x]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:94.0pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">9. feladatsor: Állandók variálása, megoldás sorfejtéssel</span></b></p> <p style="top:111.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">(megoldás)</span></b></p> <p style="top:154.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:173.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet elsőrendű lineáris, először a hozzá tartozó homogén egyenletet oldjuk</span></p> <p style="top:188.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">meg, ami szétválasztható:</span></p> <p style="top:212.3pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:228.6pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></sup></p> <p style="top:253.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A két oldal integrálása után</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagy átrendezve és</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyett</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-t írva</span></p> <p style="top:268.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik.</span></p> <p style="top:282.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenlet megoldását az állandók variálásának módszere szerint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:297.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük. Az egyenlet bal oldalába helyettesítjük:</span></p> <p style="top:324.5pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xc</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:344.4pt;left:185.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:370.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kapott kifejezésnek a jobb oldallal kell megegyeznie, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagyis</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:385.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tetszőleges konstans. Az egyenlet általános megoldása tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:399.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:419.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg az</span></p> <p style="top:452.2pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:436.3pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:444.9pt;left:142.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:444.9pt;left:172.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:459.3pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:459.3pt;left:182.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:436.3pt;left:199.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:452.2pt;left:207.3pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:436.3pt;left:228.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:444.9pt;left:234.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:459.3pt;left:250.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:436.3pt;left:271.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:485.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek</span></p> <p style="top:499.8pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldásai.</span></p> <p style="top:519.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A két függvény lineárisan független, tehát a homogén egyenlet általános meg-</span></p> <p style="top:533.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">oldása</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az a mátrix, aminek oszlopai az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:548.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektorok,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pedig az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> előtt álló együtthatómátrix. Az állandók variálásának</span></p> <p style="top:562.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">módszere szerint az inhomogén egyenlet megoldását</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban keressük.</span></p> <p style="top:577.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alapján a derivált</span></p> <p style="top:603.1pt;left:108.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:620.5pt;left:137.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:637.9pt;left:137.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:664.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> akkor oldja meg az egyenletet, ha</span></p> <p style="top:698.1pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:682.1pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:690.7pt;left:190.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:705.2pt;left:206.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:682.1pt;left:227.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:731.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">teljesül. Számítsuk ki</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> inverzét:</span></p> <p style="top:767.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:751.9pt;left:158.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:760.5pt;left:166.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:760.5pt;left:193.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:774.9pt;left:164.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:774.9pt;left:189.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:751.9pt;left:204.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:767.8pt;left:224.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p 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style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:774.9pt;left:403.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:774.9pt;left:430.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:751.9pt;left:460.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:767.8pt;left:468.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> </div>	page_283.png	Matematika A3 gyakorlat Energetika és Mechatzonika BsSc szakok, 2016/17 ősz 9. feladatsor: Állandók variálása, megoldás sorfejtéssel (megoldás) 1. Határozzuk meg az y/ - 2zy — 22€77 differenciálegyenlet általános megoldását. "Megoldás. Az egyenlet elsőrendű lineáris, először a hozzá tartozó homogén egyenletet oldjuk meg, ami szétválasztható: A két oldal integrálása után I [y(z)] — €, vagy átrendezve és 2e" helyett C-t írva we) — Ce-" adódik. Az inhomogén egyet elzje "" alakban keressük. Az egyenlet bal oldalába helyettesítjűk: let. megoldását az állandók variálásának módszere szerint y(z) — aE — elejaze ? 4 2e eeer v(z) 4 2xvíz) A kapott kifejezésnek a jobb oldallal kell megegyeznie, tehát el(r) — 2r. vagyis elz) wla) te 224.C, ahol C tetszőleges konstans. Az egyenlet általános megoldása t. bc 2. Határozzuk meg az oldása yíz) — Cyyi(r) 4 Cgyalr). Legyen Ulr) az a mátrix, aminek oszlopai az y.(r) KÖLGETTŐTŐI A(rjU(ejelz) 4 Uizjelz) A(ryy(z) 4 Ulejela). v voz-vo[] v [fl ,É]d
ezzel " # −e−x 2e−3x 1 −e−2x " −6e−2x# c′(x) = " # 6e−3x . −6e−2x Integráljuk a kapott vektort (komponensenként), majd szorozzuk meg az U(x) mátrixszal: " # −2e−3x + C1 3e−2x + C2 " 4e−2x U(x) · c(x) = " ex 2 e3x e2x + U(x) · " # . C1 C2 Az inhomogén egyenlet általános megoldása y(x) = (4e−2x, 1) + C1y1(x) + C2y2(x). 3. Határozzuk meg az y′ = 1+x2 0 1 1 1+x2 y + 1+x2 diﬀerenciálegyenlet-rendszer általános megoldását. Megoldás. Először a homogén egyenletrendszert oldjuk meg. Írjuk fel az komponensenként: y′ 1 = 1 1 + x2y1 y′ 2 = y1 + 1 1 + x2y2. Ebben az alakban láthatjuk, hogy az első egyenlet nem tartalmazza y2-t, így abból y1 elvileg meghatározható. Ez elsőrendű homogén lineáris, tehát szétválasztható: y′ 1 y1 1 1 + x2, aminek a megoldása y1(x) = Cearctan x. Most a második egyenletbe írjuk be a kapott függvényt (pl. C = 1 választással), ami így elsőrendű inhomogén lineáris lesz. A homogén rész megegyezik az imént megoldott egyenlettel, tehát az inhomogén egyenlet megoldását y2(x) = c(x)earctan x alakban kereshetjük. Behelyettesítve a következő egyenlet adódik: c′(x)earctan x + c(x)earctan x 1 + x2 = earctan x + 1 1 + x2c(x)earctan x, amiből c′(x) = 1, azaz c(x) = x + C. Az eddigiek alapján felírhatjuk a homogén egyenlet általános megoldását: " # earctan x xearctan x " # earctan x 0 xearctan x earctan x {z U(x) yh(x) = C1 + C2 earctan x " # . C1 C2 Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása U(x)c(x), ahol c′(x) = U(x)−1 · = e−arctan x 1+x2 1 " 1 −x 1+x2 e−arctan x " 1+x2 e−arctan x −xe−arctan x 1+x2	ezzel � _·_ �−6e[−][2][x] 0 � = � 6e[−][3][x] _−6e[−][2][x]_ � _._ c[′](x) = �−e[−][x] 2e[−][3][x] 1 _−e[−][2][x]_ Integráljuk a kapott vektort (komponensenként), majd szorozzuk meg az U (x) mátrixszal: �−2e[−][3][x] + C1� = 3e[−][2][x] + C2 � _._ � + U (x) · �C1 _C2_ _U_ (x) · c(x) = � _e[x]_ 2 � _·_ _e[3][x]_ _e[2][x]_ �4e[−][2][x] 1 Az inhomogén egyenlet általános megoldása y(x) = (4e[−][2][x], 1) + C1y1(x) + C2y2(x). 3. Határozzuk meg az � 1 1+x[2] 1 y[′] = � 1 0 1+x[2] 1 1 1+x[2] � y + � differenciálegyenlet-rendszer általános megoldását. _Megoldás. Először a homogén egyenletrendszert oldjuk meg. Írjuk fel az komponensenként:_ 1 _y1[′]_ [=] 1 + x[2] _[y][1]_ 1 _y2[′]_ [=][ y][1] [+] 1 + x[2] _[y][2][.]_ Ebben az alakban láthatjuk, hogy az első egyenlet nem tartalmazza y2-t, így abból y1 elvileg meghatározható. Ez elsőrendű homogén lineáris, tehát szétválasztható: _y1[′]_ = 1 _y1_ 1 + x[2] _[,]_ aminek a megoldása y1(x) = Ce[arctan][ x]. Most a második egyenletbe írjuk be a kapott függvényt (pl. C = 1 választással), ami így elsőrendű inhomogén lineáris lesz. A homogén rész megegyezik az imént megoldott egyenlettel, tehát az inhomogén egyenlet megoldását y2(x) = c(x)e[arctan][ x] alakban kereshetjük. Behelyettesítve a következő egyenlet adódik: 1 _c[′](x)e[arctan][ x]_ + c(x)[e][arctan][ x] 1 + x[2][ =][ e][arctan][ x][ +] 1 + x[2] _[c][(][x][)][e][arctan][ x][,]_ amiből c[′](x) = 1, azaz c(x) = x + C. Az eddigiek alapján felírhatjuk a homogén egyenlet általános megoldását: � + C2 � 0 _e[arctan][ x]_ � = � _e[arctan][ x]_ 0 _xe[arctan][ x]_ _e[arctan][ x]_ � _·_ �C1� _._ _C2_ yh(x) = C1 � _e[arctan][ x]_ _xe[arctan][ x]_ � �� _U_ (x) Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása U (x)c(x), ahol c[′](x) = U (x)[−][1] _·_ � 1 � 1+x[2] 1 � 1 0� = e[−] [arctan][ x] _·_ _−x_ 1 = _e[−]_ [arctan][ x] � 1 0� _·_ � 1+1x[2] �, _−x_ 1 1 � _e[−]_ [arctan][ x] � 1+x[2] _e[−]_ [arctan][ x] _−_ _[xe][−]_ [arctan][ x] 1+x[2] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezzel</span></p> <p style="top:87.8pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span 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style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:95.0pt;left:304.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:71.9pt;left:340.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:87.8pt;left:348.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:119.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Integráljuk a kapott vektort (komponensenként), majd szorozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mátrixszal:</span></p> <p style="top:151.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:135.4pt;left:177.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:144.0pt;left:185.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:144.0pt;left:212.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:158.5pt;left:183.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:158.5pt;left:208.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:135.4pt;left:223.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:151.4pt;left:231.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:135.4pt;left:237.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:144.0pt;left:243.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:158.5pt;left:248.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:135.4pt;left:307.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:151.4pt;left:316.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:135.4pt;left:329.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:144.0pt;left:335.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:158.5pt;left:345.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:135.4pt;left:362.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:151.4pt;left:370.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i></p> <p style="top:135.4pt;left:416.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:144.0pt;left:421.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:158.5pt;left:421.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:135.4pt;left:435.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:151.4pt;left:442.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:183.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenlet általános megoldása</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:202.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Határozzuk meg az</span></p> <p style="top:233.4pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:217.4pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:223.9pt;left:146.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:232.7pt;left:139.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:225.4pt;left:178.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:240.5pt;left:146.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:239.0pt;left:179.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:247.8pt;left:171.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:217.4pt;left:192.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:233.4pt;left:200.0pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:217.4pt;left:221.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:224.2pt;left:236.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:233.0pt;left:228.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:240.8pt;left:235.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:217.4pt;left:249.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:264.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását.</span></p> <p style="top:283.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Először a homogén egyenletrendszert oldjuk meg. Írjuk fel az komponensenként:</span></p> <p style="top:311.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:317.9pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:303.7pt;left:146.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:320.0pt;left:133.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:340.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:346.7pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:332.4pt;left:171.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:348.7pt;left:158.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:368.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebben az alakban láthatjuk, hogy az első egyenlet nem tartalmazza</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-t, így abból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> elvileg</span></p> <p style="top:383.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">meghatározható. Ez elsőrendű homogén lineáris, tehát szétválasztható:</span></p> <p style="top:405.1pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:411.2pt;left:113.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:421.4pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:413.2pt;left:122.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:405.1pt;left:149.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:421.4pt;left:136.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:443.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">aminek a megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:458.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Most a második egyenletbe írjuk be a kapott függvényt (pl.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> választással), ami így</span></p> <p style="top:472.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elsőrendű inhomogén lineáris lesz. A homogén rész megegyezik az imént megoldott egyen-</span></p> <p style="top:487.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lettel, tehát az inhomogén egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban kereshetjük.</span></p> <p style="top:501.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Behelyettesítve a következő egyenlet adódik:</span></p> <p style="top:532.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:540.2pt;left:204.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:523.9pt;left:318.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:540.2pt;left:305.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:560.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az eddigiek alapján felírhatjuk a homogén egyenlet</span></p> <p style="top:575.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldását:</span></p> <p style="top:606.4pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:590.5pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:599.1pt;left:174.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:613.6pt;left:171.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:590.5pt;left:213.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:606.4pt;left:222.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:590.5pt;left:248.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:599.1pt;left:269.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:613.6pt;left:254.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:590.5pt;left:290.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:606.4pt;left:299.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:590.5pt;left:311.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:599.1pt;left:321.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:599.1pt;left:384.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:613.6pt;left:317.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:613.6pt;left:370.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:590.5pt;left:405.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:624.2pt;left:312.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">\|</span></p> <p style="top:624.2pt;left:357.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">{z</span></p> <p style="top:624.2pt;left:407.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">}</span></p> <p style="top:636.4pt;left:352.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">U</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:606.4pt;left:413.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:590.5pt;left:418.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:599.1pt;left:424.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:613.6pt;left:424.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:590.5pt;left:437.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:606.4pt;left:445.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:656.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:670.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span></p> <p style="top:699.2pt;left:108.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:683.3pt;left:193.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:690.1pt;left:208.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:698.9pt;left:200.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:706.6pt;left:207.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:683.3pt;left:221.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:733.1pt;left:136.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:717.1pt;left:194.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:725.8pt;left:205.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:725.8pt;left:226.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:740.2pt;left:200.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:740.2pt;left:226.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:717.1pt;left:231.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:733.1pt;left:240.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:717.1pt;left:246.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:723.9pt;left:261.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:732.8pt;left:253.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:740.5pt;left:260.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:717.1pt;left:274.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:768.3pt;left:136.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:752.4pt;left:148.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:758.8pt;left:187.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> x</span></i></sup></p> <p style="top:767.7pt;left:196.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:776.7pt;left:154.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> x</span></i></sup></p> <p style="top:784.0pt;left:225.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:752.4pt;left:257.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:734.4pt;left:280.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_284.png	Az inhomogú egyenlet általános megoldása y(z) — (4e-2",1) 4 Cyilz) 4 Cayalo). Határozzuk meg az differenciálegyer 1 14 [1 11 úon tr Ebben az alakban láthatjuk, hogy az első egyenlet nem tartalmazza y.-t, így abból w, elvileg meghatározható. Ez elsőrendű homogén lincáris, tehát szótválasztható; ú 1 ml aminek a megoldása yi(r) — Cese Most a második egyenletbe írjuk be a kapott függvényt (pl. C — 1 választással), ami így elsőrendű inkomogén lincáris lesz. A homogén rész megegyezik az imént megoldott egyer lettel, tehát az inhomogén egyenlet megoldását ya(z) — elrje"snz alakban kereshetjük. Behelyettesítve a következő egyenlet adódik: elejerne 4 el2): eee 15 amiből e(z) — 1, azaz elz) — 2 4 C. Az edd általános megoldását. Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása U(z)e(r). ahol iek alapján felírhatjuk a homogén egyenlet yale) ele)
amiből c integrálással meghatározható. Ezt kell megszorozni az U(x) mátrixszal: # " earctan x 0 xearctan x earctan x U(x)c(x) = " −e−arctan x# # " −1 = . 0 xe−arctan x Az inhomogén egyenletrendszer általános megoldása tehát " earctan x xearctan x y(x) = # " −1 + C1 + C2 earctan x 4. Határozzuk meg az xy′′ −(x + 1)y′ + y = x2ex diﬀerenciálegyenlet általános megoldását, ha tudjuk, hogy y1(x) = ex és y2(x) = x + 1 megoldja a hozzá tartozó homogén egyenletet. 4. Határozzuk meg az xy′′ −(x + 1)y′ + y = x2ex diﬀerenciálegyenlet általános megoldását, ha Megoldás. Az egyenlet másodrendű lineáris, az y = (y, y′) változó bevezetésével átalakíthatjuk elsőrendű egyenletrendszerré, és alkalmazhatjuk az állandók variálásának módszerét: y′ = −1 x x+1 y + . xex ex és x + 1 lineárisan függetlenek, tehát a homogén egyenletrendszer általános megoldását " # ex x + 1 ex 1 {z U(x) " # C1 C2 alakban írhatjuk fel. Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása y(x) = c1(x)ex + c2(x)(x + 1), ahol exc′ 1 c′ 1(x) + (x + 1)c′ 2 + (x + 1)c′ 2(x) = 0 exc′ 1(x) + c′ 2(x) = x c′ 1(x) + c′ 2 c′ 2(x) = xex, amiből c′ 1 c′ 1(x) = 1 + x és c′ 2 c′ 2(x) = −ex. Integrálással kapjuk, hogy c1(x) = C1 + x + x2 2 amiből c′ 1(x) = 1 + x és c′ 2(x) = −ex. Integrálással kapjuk, hogy c1(x) = C1 + x + x2 2 és c2(x) = C2 −ex. Tehát y(x) = (C1 + x + x2 2 )ex + (x + 1)(C2 −ex). y(x) = (C1 + x + x2 2 5. Oldjuk meg sorfejtéssel az y′ = x+y diﬀerenciálegyenletet y(0) = 0 kezdeti feltétel mellett. Megoldás. A megoldást X akxk k=0 y(x) = alakban keressük, ennek deriváltja y′(x) = k=1 kakxk−1 = X (k + 1)ak+1xk. k=0 Behelyettesítve az egyenlet ∞ X (k + 1)ak+1xk = k=0 ∞ X akxk + x k=0	amiből c integrálással meghatározható. Ezt kell megszorozni az U (x) mátrixszal: �−e[−] [arctan][ x] _xe[−]_ [arctan][ x] � = �−1� _._ 0 � _·_ _U_ (x)c(x) = � _e[arctan][ x]_ 0 _xe[arctan][ x]_ _e[arctan][ x]_ Az inhomogén egyenletrendszer általános megoldása tehát � 0 _e[arctan][ x]_ � � _e[arctan][ x]_ _xe[arctan][ x]_ � + C2 y(x) = �−1� 0 + C1 . 4. Határozzuk meg az xy[′′] _−_ (x + 1)y[′] + y = x[2]e[x] differenciálegyenlet általános megoldását, ha tudjuk, hogy y1(x) = e[x] és y2(x) = x + 1 megoldja a hozzá tartozó homogén egyenletet. _Megoldás. Az egyenlet másodrendű lineáris, az y = (y, y[′]) változó bevezetésével átalakíthat-_ juk elsőrendű egyenletrendszerré, és alkalmazhatjuk az állandók variálásának módszerét: � _._ y[′] = � 0 1 � _−_ [1] _x+1_ y + _x_ _x_ � 0 _xe[x]_ _e[x]_ és x + 1 lineárisan függetlenek, tehát a homogén egyenletrendszer általános megoldását �e[x] _x + 1�_ _e[x]_ 1 � �� _U_ (x) �C1 _C2_ � alakban írhatjuk fel. Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása y(x) = c1(x)e[x] + c2(x)(x + 1), ahol _e[x]c[′]1[(][x][) + (][x][ + 1)][c][′]2[(][x][) = 0]_ _e[x]c[′]1[(][x][) +][ c][′]2[(][x][) =][ xe][x][,]_ amiből c[′]1[(][x][) = 1 +][ x][ és][ c][′]2[(][x][) =][ −][e][x][. Integrálással kapjuk, hogy][ c][1][(][x][) =][ C][1] [+][ x][ +][ x]2[2] [és] _c2(x) = C2 −_ _e[x]. Tehát_ _y(x) = (C1 + x +_ _[x][2]_ 2 [)][e][x][ + (][x][ + 1)(][C][2][ −] _[e][x][)][.]_ 5. Oldjuk meg sorfejtéssel az y[′] = x + _y differenciálegyenletet y(0) = 0 kezdeti feltétel mellett._ _Megoldás. A megoldást_ _y(x) =_ _∞_ � _akx[k]_ _k=0_ alakban keressük, ennek deriváltja _∞_ �(k + 1)ak+1x[k]. _k=0_ _y[′](x) =_ _∞_ � _kakx[k][−][1]_ = _k=1_ Behelyettesítve az egyenlet _∞_ �(k + 1)ak+1x[k] = _k=0_ _∞_ � _akx[k]_ + x _k=0_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> integrálással meghatározható. Ezt kell megszorozni az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mátrixszal:</span></p> <p style="top:93.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:77.0pt;left:168.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:85.6pt;left:178.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:85.6pt;left:241.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:100.1pt;left:174.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:100.1pt;left:226.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:77.0pt;left:262.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:93.0pt;left:271.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:77.0pt;left:277.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:85.6pt;left:282.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:100.1pt;left:284.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:77.0pt;left:335.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:93.0pt;left:344.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:77.0pt;left:357.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:85.6pt;left:363.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:100.1pt;left:367.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:77.0pt;left:378.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:93.0pt;left:386.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:126.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenletrendszer általános megoldása tehát</span></p> <p style="top:159.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:143.4pt;left:145.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:152.0pt;left:150.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:166.4pt;left:155.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:143.4pt;left:166.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:159.3pt;left:174.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:143.4pt;left:201.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:152.0pt;left:210.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:166.4pt;left:207.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:143.4pt;left:249.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:159.3pt;left:257.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:143.4pt;left:284.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:152.0pt;left:305.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:166.4pt;left:290.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:143.4pt;left:326.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:185.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:205.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet általános megoldását, ha</span></p> <p style="top:219.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tudjuk, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldja a hozzá tartozó homogén egyenletet.</span></p> <p style="top:239.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet másodrendű lineáris, az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> változó bevezetésével átalakíthat-</span></p> <p style="top:253.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">juk elsőrendű egyenletrendszerré, és alkalmazhatjuk az állandók variálásának módszerét:</span></p> <p style="top:286.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:270.7pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:279.3pt;left:143.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:279.3pt;left:170.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:293.8pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:301.1pt;left:148.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:292.3pt;left:165.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></p> <p style="top:301.1pt;left:171.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:270.7pt;left:182.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:286.7pt;left:190.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:270.7pt;left:211.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:279.3pt;left:223.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:293.8pt;left:217.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:270.7pt;left:234.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:286.7pt;left:242.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:319.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> lineárisan függetlenek, tehát a homogén egyenletrendszer általános megoldását</span></p> <p style="top:337.1pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:345.7pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:345.7pt;left:132.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></p> <p style="top:360.1pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:360.1pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:337.1pt;left:159.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:370.7pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">\|</span></p> <p style="top:370.7pt;left:131.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">{z</span></p> <p style="top:370.7pt;left:161.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">}</span></p> <p style="top:383.0pt;left:127.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">U</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:337.1pt;left:167.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:345.7pt;left:173.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:360.1pt;left:173.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:337.1pt;left:186.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:404.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban írhatjuk fel. Az állandók variálásának módszere alapján az inhomogén egyenlet</span></p> <p style="top:418.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span></p> <p style="top:444.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:450.9pt;left:122.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:450.9pt;left:198.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span></sup></p> <p style="top:462.2pt;left:142.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:468.4pt;left:158.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:468.4pt;left:198.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:490.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:497.0pt;left:121.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:497.0pt;left:211.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Integrálással kapjuk, hogy</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:498.2pt;left:516.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup></p> <p style="top:505.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Tehát</span></p> <p style="top:538.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:547.0pt;left:201.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:569.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Oldjuk meg sorfejtéssel az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:588.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megoldást</span></p> <p style="top:621.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:611.3pt;left:147.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:612.7pt;left:144.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:636.4pt;left:144.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:621.0pt;left:161.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:655.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük, ennek deriváltja</span></p> <p style="top:688.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:678.3pt;left:150.4pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:679.7pt;left:147.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:703.4pt;left:146.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:688.1pt;left:164.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:678.3pt;left:223.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:679.7pt;left:220.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:703.4pt;left:220.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:688.1pt;left:235.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:722.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Behelyettesítve az egyenlet</span></p> <p style="top:745.3pt;left:109.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:746.7pt;left:107.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:770.4pt;left:106.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:755.1pt;left:121.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:745.3pt;left:210.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:746.7pt;left:207.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:770.4pt;left:207.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:755.1pt;left:224.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_285.png	amiből e integrálással meghatározható. Ezt kell megszorozni az (r) mátrisszal: v :[n'] . [;;;mí] .. [?] Határozzuk meg az 14/— (r-4 1V 4-y — 2e" díflerenciá tudjuk, hogy ylz) — e és yalr; Uirjelz) egyenlet általános megoldását, ha. 4. 1 megoldja a hozzá tartozó homogén egyenletet. áris, az y — (y. ) változó bevezetésével átalakíthat- és alkalmazhatjuk az állandók variálásának módszerú € és r. 1 lncárisan fűggetlenek, tehát a homogén egyenletrendszer általános megoldását e 21] [1 e 1 [le megoldása y(z) — eilz)e? 4-eslz)(r 4-1), ahol edlet(esNélr FA eze, amiből d(z) — 14 7 és élr ealz) — C — €". Tehát e. Intográlással kapjuk, hogy el2) — C1 434 £ és v l ró '7 e 4 (r 4 (C — €). Oldjuk meg sorfejtéssel az y/ — £ y difforenciálegyenletet y(0) — 0 kezdeti feltétel mellett .Megoldás. A megoldást alakban keressük, ennek deríváltja v AÉLMH íllm Va Behelyettesítve az egyenlet
alakú lesz, amiből az együtthatók összehasonlításával ak+1 = ak k+1 ha k ̸= 1 a1+1 ha k = 1 adódik. A kezdeti feltétel 0 = y(0) = a0, tehát 0 ha k ≤1 1 k! egyébként ak = A sor minden x-re abszolút konvergens, összegfüggvénye y(x) = ex −1 −x, ami valóban megoldás. További gyakorló feladatok 6. Határozzuk meg az y′ −(tan x + ctg x)y = −4 sin2 x diﬀerenciálegyenlet általános megoldá- sát. Megoldás. Az egyenlet elsőrendű lineáris inhomogén, először a homogén egyenletet oldjuk meg. Ez szétválasztható: y′ y = tan x + ctg x. Integráljuk mindkét oldalt, ebből a homogén egyenlet általános megoldása Z sin x ln \|yh(x)\| = sin x cos x + cos x sin x sin x dx = −ln cos x + ln sin x + C = ln tan x + C, vagyis yh(x) = C tan x. Az inhomogén egyenlet megoldható az állandók variálásával. y(x) = c(x)yh(x), ahol c′(x) = −4 sin2 x tan x = −4 sin x cos x = −2 sin 2x. Ennek az integrálja c(x) = cos 2x+C, tehát az eredeti egyenlet általános megoldása y(x) = cos 2x tan x + C tan x. 7. Oldjuk meg az xy′ −y = x3 + 1 diﬀerenciálegyenletet y(2) = 5 kezdeti feltétel mellett. Megoldás. Az egyenlet elsőrendű lineáris inhomogén, a hozzá tartozó homogén egyenlet szétválasztható: y′ y′ y = 1 x 1 x, integrálás után az yh(x) = Cx általános megoldást kapjuk. Az inhomogén egyenlet megoldását az állandók variálásának módszerével keressük. y(x) = c(x)x, a kezdeti feltételből c(2) = 5 x2 , tehát 2, az egyenletből pedig c′(x) = x3+1 x2 5 2, az egyenletből pedig c′(x) = x3+1 3+1 x2 , tehát Z x ξ3 + 1 c(x) = 5 2 + c(x) = 5 2 2 + 1 ξ2 dξ = 5 5 2 −3 2 3 2 −1 x 2 1 x + x2 x 2 . A kezdetiérték-probléma megoldása y(x) = −1 + x + x3 2 x3 2 . 8. Határozzuk meg az y′ + y = e−x diﬀerenciálegyenlet általános megoldását.	alakú lesz, amiből az együtthatók összehasonlításával _ak+1 =_    _ak_ ha k ̸= 1 _k+1_ _a1+1_ ha k = 1 2 adódik. A kezdeti feltétel 0 = y(0) = a0, tehát _ak =_  0 ha k ≤ 1 1 egyébként  _k!_ A sor minden x-re abszolút konvergens, összegfüggvénye y(x) = e[x] _−_ 1 − _x, ami valóban_ megoldás. ## További gyakorló feladatok 6. Határozzuk meg az y[′] _−_ (tan x + ctg x)y = −4 sin[2] _x differenciálegyenlet általános megoldá-_ sát. _Megoldás. Az egyenlet elsőrendű lineáris inhomogén, először a homogén egyenletet oldjuk_ meg. Ez szétválasztható: _y[′]_ _y_ [= tan][ x][ + ctg][ x.] Integráljuk mindkét oldalt, ebből a homogén egyenlet általános megoldása �� sin x ln \|yh(x)\| = cos x [+ cos]sin x[ x] � dx = − ln cos x + ln sin x + C = ln tan x + C, vagyis yh(x) = C tan x. Az inhomogén egyenlet megoldható az állandók variálásával. y(x) = c(x)yh(x), ahol _c[′](x) =_ _[−][4 sin][2][ x]_ = −4 sin x cos x = −2 sin 2x. tan x Ennek az integrálja c(x) = cos 2x + _C, tehát az eredeti egyenlet általános megoldása y(x) =_ cos 2x tan x + C tan x. 7. Oldjuk meg az xy[′] _−_ _y = x[3]_ + 1 differenciálegyenletet y(2) = 5 kezdeti feltétel mellett. _Megoldás. Az egyenlet elsőrendű lineáris inhomogén, a hozzá tartozó homogén egyenlet_ szétválasztható: _yy[′]_ [= 1]x _[,]_ integrálás után az yh(x) = Cx általános megoldást kapjuk. Az inhomogén egyenlet megoldását az állandók variálásának módszerével keressük. y(x) = _c(x)x, a kezdeti feltételből c(2) =_ [5] 2 [, az egyenletből pedig][ c][′][(][x][) =][ x][3]x[+1][2][, tehát] � _x_ _ξ[3]_ + 1 _c(x) = [5]_ dξ = [5] 2 [+] 2 _ξ[2]_ 2 _[−]_ [3]2 _[−]_ _x[1]_ [+][ x]2[2] _[.]_ A kezdetiérték-probléma megoldása y(x) = −1 + x + _[x][3]_ 2 [.] 8. Határozzuk meg az y[′] + y = e[−][x] differenciálegyenlet általános megoldását. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú lesz, amiből az együtthatók összehasonlításával</span></p> <p style="top:95.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:75.9pt;left:144.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:84.8pt;left:144.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:102.8pt;left:144.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:85.2pt;left:157.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">k</span></i></p> <p style="top:94.3pt;left:154.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></p> <p style="top:87.0pt;left:186.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></p> <p style="top:102.8pt;left:154.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></p> <p style="top:111.7pt;left:161.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:104.3pt;left:186.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></p> <p style="top:132.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik. A kezdeti feltétel</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:168.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:149.4pt;left:133.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:158.4pt;left:133.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:176.3pt;left:133.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:160.6pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:160.6pt;left:163.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:176.4pt;left:144.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:185.2pt;left:143.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">!</span></p> <p style="top:177.9pt;left:163.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyébként</span></p> <p style="top:205.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A sor minden</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re abszolút konvergens, összegfüggvénye</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami valóban</span></p> <p style="top:220.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás.</span></p> <p style="top:252.7pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:276.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ctg</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldá-</span></p> <p style="top:291.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sát.</span></p> <p style="top:310.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet elsőrendű lineáris inhomogén, először a homogén egyenletet oldjuk</span></p> <p style="top:325.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">meg. Ez szétválasztható:</span></p> <p style="top:349.0pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:365.3pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= tan</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ctg</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></sup></p> <p style="top:388.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Integráljuk mindkét oldalt, ebből a homogén egyenlet általános megoldása</span></p> <p style="top:419.6pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:407.1pt;left:167.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z </span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:427.8pt;left:188.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:427.8pt;left:229.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:406.6pt;left:254.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:419.6pt;left:263.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ln sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = ln tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C,</span></i></p> <p style="top:450.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vagyis</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:464.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenlet megoldható az állandók variálásával.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span></p> <p style="top:498.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 sin</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:506.5pt;left:156.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:498.2pt;left:196.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4 sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></p> <p style="top:528.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ennek az integrálja</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát az eredeti egyenlet általános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:542.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:562.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2) = 5</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:581.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet elsőrendű lineáris inhomogén, a hozzá tartozó homogén egyenlet</span></p> <p style="top:596.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szétválasztható:</span></p> <p style="top:617.6pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:633.9pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></sup></p> <p style="top:633.9pt;left:134.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:658.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">integrálás után az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> általános megoldást kapjuk.</span></p> <p style="top:672.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenlet megoldását az állandók variálásának módszerével keressük.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:686.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a kezdeti feltételből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(2) =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup></p> <p style="top:694.2pt;left:252.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az egyenletből pedig</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></sup></p> <p style="top:694.2pt;left:419.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> , tehát</span></sup></p> <p style="top:721.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></sup></p> <p style="top:729.7pt;left:144.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:709.0pt;left:165.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:733.2pt;left:171.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:713.4pt;left:184.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span></p> <p style="top:729.7pt;left:194.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:721.5pt;left:217.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></sup></p> <p style="top:729.7pt;left:247.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup></p> <p style="top:729.7pt;left:269.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:729.7pt;left:292.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:729.7pt;left:319.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:756.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdetiérték-probléma megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup></p> <p style="top:763.6pt;left:355.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:775.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet általános megoldását.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_286.png	sz, amiből az együtthatók összehasonlításával oéx hakzi mAz$lh pak adódik, A kezdeti feltétel 0 — 9(0) — a tehét MORTTET " 78 esyébként A sor minden 2-re abszolút könvergens, összegfűggi e — 1— x, ami valóban megoldás. "További gyakorló feladatok sát. álogyenlet általános megoldá- .Megoldás. Az egyenlet elsőrendü lineáris inhomogén, először a homogén egyet meg, Ez szétválasztható: letet oldjuk Y m tanz 4 etsz. v Integráljuk mindi ml / (Z ét oldalt, ebből a homogén egyenlet általános megoldása. Incosz 4 Insinz 4 C — lntanz 4 €, 4sinreosz — —2sin2r Ennek az integrálja elz) — cos2r4-C, tehát az eredeti egyenlet általános megoldása y(z) — cos2ztanr 4 Ctanr 7. Oldjuk meg az 2y/ — y — 2941 díffer iálogye 5 kezdeti feltétel mellett. letet y(2) .Megoldás. Az egyenlet elsörendű lincáris inhomogén, a hozzá tartozó homogé egye szétválasztható: val vőz integrálás után az ya(r) — Cr általános megoldást kapjuk. Az inhomogén egyenlet megoldását az állandók variálásának módszerével keressük. y(z) — elzje, a kezdeti feltételből e(2) — §. az egyenletből pedig e(z) — 2221. tehát 5 , fÉrI 5 3 1 - docAu to ; A kezdetiértél 8. Határozzuk meg az y/ 4. y — €-? dífforenciálegyenlet általános megoldását. 4
Megoldás. Az egyenlet elsőrendű lineáris inhomogén, a hozzá tartozó homogén egyenlet y′ + y = 0, ennek megoldása ln \|y(x)\| = (−1) dx = −x + C, azaz yh(x) = Ce−x. Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása y(x) = c(x)e−x, ahol c′(x) = e−x e−x = 1, tehát c(x) = x + C. Az általános megoldás y(x) = xe−x + Ce−x. 9. Oldjuk meg az y′ + y cos x = sin x cos x diﬀerenciálegyenletet y(0) = 1 kezdeti feltétel mellett. Megoldás. Az egyenlet elsőrendű lineáris inhomogén, a homogén egyenlet átrendezve y′ y = −cos x. Ebből a homogén egyenlet általános megoldása Ce−sin x. Az inhomogén egyenlet megoldása y(x) = c(x)e−sin x alakba írható, ahol c(0) = 1 és c′(x) = sin x cos x e−sin x , tehát (t = sin x helyettesítéssel, majd parciálás integrálással) Z x Z x 0 esin ξ sin ξ cos ξ dξ c(x) = 1 + = 1 + 0 Z sin x tet dt 0 = 1 + [ueu −eu]u=sin x u=0 = 2 + esin x sin x −esin x. 0 = 1 + [ueu −eu]u=sin x u=0 A kezdetiérték-probléma megoldása tehát y(x) = −1 + 2e−sin x + sin x. 10. Határozzuk meg az y′ = 1 −x2 −x " # y + x 1 diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (1 + x2, x) és y2(x) = (x, 1) megoldásai. Megoldás. A megadott függvények lineárisan függetlenek, legyen U(x) = " # 1 + x2 x a belőlük képzett mátrix. Az állandók variálásának módszere szerint az általános megoldás y(x) = U(x)c(x), ahol a c(x) függvényre c′(x) = U(x)−1 " # " # " # · = x 1 0 1 1 −x −x 1 + x2 teljesül. Ennek primitív függvénye c(x) = (C1, x + C2), tehát az általános megoldás " 1 + x2 " x2# y(x) = " # C1 x + C2 + C1y1(x) + C2y2(x).	_Megoldás. Az egyenlet elsőrendű lineáris inhomogén, a hozzá tartozó homogén egyenlet_ _y[′]_ + y = 0, ennek megoldása � ln \|y(x)\| = (−1) dx = −x + C, azaz yh(x) = Ce[−][x]. Az állandók variálásának módszere alapján az inhomogén egyenlet megoldása y(x) = c(x)e[−][x], ahol _c[′](x) =_ _[e][−][x]_ _e[−][x][ = 1][,]_ tehát c(x) = x + C. Az általános megoldás y(x) = xe[−][x] + Ce[−][x]. 9. Oldjuk meg az y[′] + y cos x = sin x cos x differenciálegyenletet y(0) = 1 kezdeti feltétel mellett. _Megoldás. Az egyenlet elsőrendű lineáris inhomogén, a homogén egyenlet átrendezve_ _y[′]_ _y_ [=][ −] [cos][ x.] Ebből a homogén egyenlet általános megoldása Ce[−] [sin][ x]. Az inhomogén egyenlet megoldása _y(x) = c(x)e[−]_ [sin][ x] alakba írható, ahol c(0) = 1 és _c[′](x) = [sin][ x][ cos][ x]_ _,_ _e[−]_ [sin][ x] tehát (t = sin x helyettesítéssel, majd parciálás integrálással) � _x_ _c(x) = 1 +_ 0 _[e][sin][ ξ][ sin][ ξ][ cos][ ξ][ d][ξ]_ � sin x = 1 + _te[t]_ dt 0 = 1 + [ue[u] _−_ _e[u]][u]u[=sin]=0_ _[ x]_ = 2 + e[sin][ x] sin x − _e[sin][ x]._ A kezdetiérték-probléma megoldása tehát y(x) = −1 + 2e[−] [sin][ x] + sin x. 10. Határozzuk meg az � y + �x 1 y[′] = �x 1 − _x[2]_ 1 _−x_ � differenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek y1(x) = (1 + x[2], x) és y2(x) = (x, 1) megoldásai. _Megoldás. A megadott függvények lineárisan függetlenek, legyen_ _U_ (x) = �1 + x[2] _x�_ _x_ 1 a belőlük képzett mátrix. Az állandók variálásának módszere szerint az általános megoldás y(x) = U (x)c(x), ahol a c(x) függvényre �x� c[′](x) = U (x)[−][1] = 1 � 1 _−x_ _−x_ 1 + x[2] � _·_ �x 1 � = �0� 1 teljesül. Ennek primitív függvénye c(x) = (C1, x + C2), tehát az általános megoldás � _C1_ _x + C2_ � = �x[2] _x_ � + C1y1(x) + C2y2(x). y(x) = �1 + x[2] _x�_ _·_ _x_ 1 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet elsőrendű lineáris inhomogén, a hozzá tartozó homogén egyenlet</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ennek megoldása</span></p> <p style="top:99.1pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:86.6pt;left:162.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:99.1pt;left:172.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C,</span></i></p> <p style="top:125.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az állandók variálásának módszere alapján az inhomogén egyenlet</span></p> <p style="top:140.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span></p> <p style="top:169.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:177.6pt;left:147.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:195.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:213.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:228.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett.</span></p> <p style="top:246.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet elsőrendű lineáris inhomogén, a homogén egyenlet átrendezve</span></p> <p style="top:266.3pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:282.6pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></sup></p> <p style="top:303.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből a homogén egyenlet általános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az inhomogén egyenlet megoldása</span></p> <p style="top:318.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakba írható, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></p> <p style="top:345.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:354.0pt;left:156.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:345.8pt;left:197.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:372.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítéssel, majd parciálás integrálással)</span></p> <p style="top:398.6pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span></p> <p style="top:386.2pt;left:165.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:410.4pt;left:170.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ξ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ξ</span></i></sup></p> <p style="top:427.3pt;left:132.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1 +</span></p> <p style="top:414.8pt;left:165.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:439.0pt;left:170.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:427.3pt;left:194.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">te</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></p> <p style="top:450.8pt;left:132.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1 + [</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ue</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:457.4pt;left:214.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:468.2pt;left:132.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:491.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdetiérték-probléma megoldása tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:509.0pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Határozzuk meg az</span></p> <p style="top:538.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:522.5pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:531.1pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:531.1pt;left:154.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:545.6pt;left:138.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:545.6pt;left:162.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:522.5pt;left:186.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:538.5pt;left:194.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:522.5pt;left:215.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:531.1pt;left:221.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:545.6pt;left:222.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:522.5pt;left:228.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:567.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását, ha a hozzá tartozó homogén rendszernek</span></p> <p style="top:582.3pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldásai.</span></p> <p style="top:600.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megadott függvények lineárisan függetlenek, legyen</span></p> <p style="top:629.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:613.6pt;left:147.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:622.2pt;left:153.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:622.2pt;left:194.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:636.7pt;left:165.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:636.7pt;left:195.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:613.6pt;left:201.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:658.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a belőlük képzett mátrix. Az állandók variálásának módszere szerint az általános megoldás</span></p> <p style="top:673.4pt;left:77.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol a</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> c</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvényre</span></p> <p style="top:703.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:687.5pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:696.1pt;left:190.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:710.6pt;left:191.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:687.5pt;left:197.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:703.5pt;left:302.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:687.5pt;left:315.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:696.1pt;left:326.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:696.1pt;left:354.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:710.6pt;left:320.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:710.6pt;left:346.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:687.5pt;left:378.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:703.5pt;left:387.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:687.5pt;left:393.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:696.1pt;left:398.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:710.6pt;left:399.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:687.5pt;left:405.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:703.5pt;left:414.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:687.5pt;left:427.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:696.1pt;left:432.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:710.6pt;left:432.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:687.5pt;left:438.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:733.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">teljesül. Ennek primitív függvénye</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát az általános megoldás</span></p> <p style="top:763.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:747.7pt;left:145.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:756.4pt;left:150.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:756.4pt;left:192.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:770.8pt;left:163.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:770.8pt;left:192.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:747.7pt;left:199.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:763.7pt;left:207.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:747.7pt;left:213.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:756.4pt;left:230.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:770.8pt;left:219.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:747.7pt;left:253.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:763.7pt;left:327.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:747.7pt;left:339.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:756.4pt;left:345.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:770.8pt;left:347.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:747.7pt;left:356.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:763.7pt;left:365.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_287.png	10. .Megoldás. Az egyenlet elsőrendű lincáris inhomogén, a hozzá tartozó homogén egyenlet W 4-y — 0. ennek megoldása tiylól- fydso ss azaz íz) — megoldása y(z) — elz)e Az állandók variálásának módszere alapján az inhomogén egyenlet -, ahol LGI tehát elr) — x. C. Az általános megoldás y(z) — ze-: 4 Ce-t. Oldjuk meg az y 4 yeosz mellett. sinzcosz differenciálegyenletet y(0) — 1 kezdeti feltétel . Megoldás. Az egyenlet elsőn ís inhomogén, a homogén egyenlet átrendezve egyenlet általános megoldása Ce-t32, Az inhomogén egyenlet megoldása alakba írható, ahol c0) — - tehát (t — sin.z helyettesi sel, majd parciálás integrálással) edoje13 Á'me.s.mgaz 214h A kezdetié éleprobléma megoldása tehát y(r) — —1 4. Határozzuk meg az n— 14-2.2) és yelr) v - [' Fa ',] a belőlűk képzett mátrix. Az állandók variálásának módszere szerint az általános megoldás y(e) — Utzjela), ahol a elz) füs 90-ven] :E -] teljesül. Ennek primitív fűggvénye elz) — (C1.2 4 C), tehát az általános megoldás o- ] [ő] - [tano cn iltalános megoldását, ha a hozzá tartozó homo (z.1) megoldásai. etlenek, legyen
11. Oldjuk meg az y′ = 1+x2 x √ 1+x2 1+x2 x2 √ 1+x2 1+x2 y + diﬀerenciálegyenlet-rendszert y(0) = (0, 1) kezdeti feltétel mellett. Megoldás. Az egyenlethez tartozó homogén egyenlet komponensenként felírva y′ 1 = x 1 + x2y1 y′ 2 = x √ 1 + x2y1 + x 1 + x2y2, az első egyenletben nincsen y2, tehát abból y1 meghatározható: y′ 1 x y1 1 + x2 mindkét oldalát integrálva ln \|y1(x)\| = 1 2 1 2 ln(1 + x2) + C, azaz y1(x) = C tb kk √ mindkét oldalát integrálva ln \|y1(x)\| = 1 2 ln(1 + x2) + C, azaz y1(x) = C 1 + x2. C = 1 választással írjuk be a második egyenletbe, ekkor az y′ 2 = x + x 1 + x2y2 y′ 2 √ √ inhomogén egyenlethez jutunk, a hozzá tartozó homogén egyenlet megegyezik az előzővel, tehát megoldása C 1 + x2 1 + x2. Az inhomogén egyenlet megoldását y2(x) = c(x) √ √ vel, tehát megoldása C 1 + x2. Az inhomogén egyenlet megoldását y2(x) = c(x) 1 + x2 alakban írjuk fel, ahol √ 1 + x2. Az inhomogén egyenlet megoldását y2(x) = c(x) c′(x) = x √ 1 + x2, tehát c(x) = √ 1 + x2 + C és így y2(x) = 1 + x2 + C √ √ tehát c(x) = 1 + x2 + C és így y2(x) = 1 + x2 + C 1 + x2. Az eddigiek alapján felírható a homogén egyenletrendszer általános megoldása: "√ 1 + x2 1 + x2 "√ yh(x) = C1 + C2 √ 1 + x2 = U(x) # . C1 C2 Az inhomogén egyenletet az állandók variálásával oldjuk meg: y(x) = U(x)c(x), ahol c′(x) = U(x)−1 · 1+x2 x2 √ 1+x2 √ 1+x2 −1 √ 1+x2 1+x2 x2 √ 1+x2 # (1+x2)3/2 −1 Az első komponens integrálásához x = sinh t, dx = cosh t dt helyettesítést alkalmazunk: 1 (1 + x2)3/2 dx = 1 cosh3 t cosh t dt = 1 cosh2 t dt = tanh t = x √ 1 + x2 + C1. A második komponens primitív függvénye −x + C2. Az inhomogén egyenlet általános megoldását az U(x) mátrixszal szorzás adja: y(x) = + C1 "√ 1 + x2 1 + x2 + C2 √ 1 + x2 a kezdeti feltételből C1 = 0 és C2 = 1. 12. Határozzuk meg az y′′ −y = 2 x Határozzuk meg az y′′ −y = 2 x −2x ln x diﬀerenciálegyenlet általános megoldását (x > 0), ha tudjuk, hogy y1(x) = ex és y2(x) = e−x megoldja a hozzá tartozó homogén egyenletet.	11. Oldjuk meg az � 1 � 1+x[2] _−_ _√_ _x[2]_ 1+x[2] y[′] = � _x_ 0 1+x[2] _√_ _x_ _x_ 1+x[2] 1+x[2] � y + differenciálegyenlet-rendszert y(0) = (0, 1) kezdeti feltétel mellett. _Megoldás. Az egyenlethez tartozó homogén egyenlet komponensenként felírva_ _x_ _y1[′]_ [=] 1 + x[2] _[y][1]_ _x_ _x_ _y2[′]_ [=] _√_ 1 + x[2] _[y][1][ +]_ 1 + x[2] _[y][2][,]_ az első egyenletben nincsen y2, tehát abból y1 meghatározható: _y1[′]_ = _x_ _y1_ 1 + x[2] _√_ mindkét oldalát integrálva ln \|y1(x)\| = [1]2 [ln(1 +][ x][2][) +][ C][, azaz][ y][1][(][x][) =][ C] 1 + x[2]. C = 1 választással írjuk be a második egyenletbe, ekkor az _x_ _y2[′]_ [=][ x][ +] 1 + x[2] _[y][2]_ inhomogén egyenlethez jutunk, a hozzá tartozó homogén egyenlet megegyezik az előző_√_ _√_ vel, tehát megoldása C 1 + x[2]. Az inhomogén egyenlet megoldását y2(x) = c(x) 1 + x[2] alakban írjuk fel, ahol _x_ _c[′](x) =_ _√_ 1 + x[2] _[,]_ _√_ _√_ tehát c(x) = 1 + x[2] + C és így y2(x) = 1 + x[2] + C 1 + x[2]. Az eddigiek alapján felírható a homogén egyenletrendszer általános megoldása: � 0 _√_ 1 + x[2] � = U (x) �C1 _C2_ � _._ yh(x) = C1 _√_ � 1 + x[2] 1 + x[2] � + C2 Az inhomogén egyenletet az állandók variálásával oldjuk meg: y(x) = U (x)c(x), ahol � _√_ 1 0 1+x[2] _−1_ _√_ 1 1+x[2] � _·_ � 1 1+x[2] _−_ _√_ _x[2]_ 1+x[2] � = � 1 (1+x[2])[3][/][2] _−1_ c[′](x) = U (x)[−][1] _·_ � 1 1+x[2] _−_ _√_ _x[2]_ 1+x[2] � = � Az első komponens integrálásához x = sinh t, dx = cosh t dt helyettesítést alkalmazunk: � 1 � 1 � 1 _x_ _√_ (1 + x[2])[3][/][2][ d][x][ =] cosh[3] _t_ [cosh][ t][ d][t][ =] cosh[2] _t_ [d][t][ = tanh][ t][ =] 1 + x[2][ +][ C][1][.] A második komponens primitív függvénye −x + C2. Az inhomogén egyenlet általános megoldását az U (x) mátrixszal szorzás adja: � � 0 + C2 _√1 + x[2]_ _√_ � � 1 + x[2] + C1 1 + x[2] � _,_ y(x) = �x 0 a kezdeti feltételből C1 = 0 és C2 = 1. 12. Határozzuk meg az y[′′] _−_ _y =_ [2] _x_ _[−]_ [2][x][ ln][ x][ differenciálegyenlet általános megoldását (][x >][ 0),] ha tudjuk, hogy y1(x) = e[x] és y2(x) = e[−][x] megoldja a hozzá tartozó homogén egyenletet. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Oldjuk meg az</span></p> <p style="top:92.4pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:76.5pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:82.5pt;left:150.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:91.3pt;left:142.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:84.0pt;left:185.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:97.5pt;left:150.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:100.6pt;left:139.1pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:107.3pt;left:146.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:97.5pt;left:185.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:106.4pt;left:178.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:76.5pt;left:199.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:92.4pt;left:207.0pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:76.5pt;left:228.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:81.9pt;left:251.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:90.7pt;left:243.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:99.7pt;left:234.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:98.2pt;left:253.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:101.2pt;left:244.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:107.9pt;left:251.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:76.5pt;left:272.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:126.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:145.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlethez tartozó homogén egyenlet komponensenként felírva</span></p> <p style="top:172.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:179.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:164.7pt;left:146.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:181.0pt;left:133.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:199.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:205.3pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:191.1pt;left:151.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:198.5pt;left:133.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:208.4pt;left:143.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:191.1pt;left:215.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:207.4pt;left:202.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:230.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az első egyenletben nincsen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát abból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> meghatározható:</span></p> <p style="top:253.0pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:259.2pt;left:113.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:269.3pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:261.1pt;left:122.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:253.0pt;left:148.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:269.3pt;left:136.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:294.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mindkét oldalát integrálva</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:301.6pt;left:281.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln(1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, azaz</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup></p> <p style="top:284.5pt;left:453.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:294.3pt;left:463.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></p> <p style="top:308.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">választással írjuk be a második egyenletbe, ekkor az</span></p> <p style="top:336.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:342.3pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:328.1pt;left:167.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:344.3pt;left:154.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:365.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">inhomogén egyenlethez jutunk, a hozzá tartozó homogén egyenlet megegyezik az előző-</span></p> <p style="top:379.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vel, tehát megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></p> <p style="top:370.0pt;left:196.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:379.8pt;left:206.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az inhomogén egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:370.0pt;left:497.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:379.8pt;left:506.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:394.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban írjuk fel, ahol</span></p> <p style="top:421.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:413.6pt;left:164.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:421.0pt;left:147.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:430.8pt;left:156.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:455.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:445.3pt;left:144.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:455.1pt;left:154.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és így</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></p> <p style="top:445.3pt;left:341.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:455.1pt;left:351.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az eddigiek alapján felírható</span></p> <p style="top:469.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a homogén egyenletrendszer általános megoldása:</span></p> <p style="top:501.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:485.8pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:494.7pt;left:181.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:509.1pt;left:176.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:485.8pt;left:213.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:501.7pt;left:221.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:485.8pt;left:248.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:494.0pt;left:272.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:499.3pt;left:254.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:509.1pt;left:264.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:485.8pt;left:295.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:501.7pt;left:304.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:485.8pt;left:344.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:494.4pt;left:350.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:508.8pt;left:350.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:485.8pt;left:363.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:501.7pt;left:370.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:534.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenletet az állandók variálásával oldjuk meg:</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span></p> <p style="top:568.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:552.5pt;left:191.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:557.9pt;left:214.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:566.7pt;left:206.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:575.7pt;left:197.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:574.2pt;left:216.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:577.3pt;left:207.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:584.0pt;left:215.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:552.5pt;left:235.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:568.5pt;left:245.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:552.5pt;left:257.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:558.0pt;left:275.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:561.1pt;left:264.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:567.8pt;left:271.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:559.6pt;left:314.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:575.6pt;left:270.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:574.1pt;left:314.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:577.1pt;left:303.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:583.8pt;left:310.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:552.5pt;left:331.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:568.5pt;left:340.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:552.5pt;left:346.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:557.9pt;left:369.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:566.7pt;left:361.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:575.7pt;left:351.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:574.2pt;left:371.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:577.3pt;left:362.4pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:584.0pt;left:369.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:552.5pt;left:390.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:568.5pt;left:399.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:552.5pt;left:411.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:558.6pt;left:435.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:567.7pt;left:418.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(1+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:576.7pt;left:430.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:552.5pt;left:458.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:601.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az első komponens integrálásához</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = cosh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítést alkalmazunk:</span></p> <p style="top:619.2pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:623.6pt;left:143.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:639.9pt;left:119.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:619.2pt;left:205.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:623.6pt;left:232.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:640.4pt;left:218.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:619.2pt;left:312.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:623.6pt;left:338.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:640.4pt;left:325.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = tanh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:623.6pt;left:452.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:631.0pt;left:434.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:640.9pt;left:444.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:663.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A második komponens primitív függvénye</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:663.0pt;left:365.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az inhomogén egyenlet általános</span></p> <p style="top:677.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldását az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mátrixszal szorzás adja:</span></p> <p style="top:710.2pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:694.2pt;left:145.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:702.9pt;left:150.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:717.3pt;left:151.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:694.2pt;left:157.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:710.2pt;left:166.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:694.2pt;left:192.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:703.2pt;left:208.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:717.6pt;left:203.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:694.2pt;left:240.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:710.2pt;left:248.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:694.2pt;left:275.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:702.5pt;left:299.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:707.8pt;left:281.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:717.6pt;left:291.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:694.2pt;left:323.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:710.2pt;left:330.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:742.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a kezdeti feltételből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:761.2pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:768.5pt;left:228.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x ></span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">),</span></sup></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha tudjuk, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldja a hozzá tartozó homogén egyenletet.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> </div>	page_288.png	1. 12. Oldjuk meg az differenciálegyer dszert y(0) — (0,1) kezdeti feltétel mellett thez tartozó homogén egyenlet kömponensenként felírva .Megoldás. Az egyen ml mindkét oldalát integrálva I [yn(2)] — $1n(14-z?) 4 C, azaz m(z) — CYTT C — 1 választással írjuk be a második egyenletbe, ekkor az TE é thez jutunk, a hozzá tartozó homogén egyenlet megegyezik az előző- t megoldása CT 473. Az inhomogén egyenlet megoldását ya(z) — elzjVI 377 alakban írjuk fel, ahol MEr tehát elz) a VTTE 4 C és így yola) — 142 4 CYT TE. Az eddígiek alapján felírható egyenletet az állandók variálásával oldjuk meg: y(z) Az első komponens integrálásához x — sinh !, dr — cosh t df helyettesítést alkalmaztnk: / i;,mü J sr kt f t tánbt T t A második kömponens primitív függyénye —z 4. Cy. Az inhomogén egyenlet általános megoldását az U(z) mátrixszal szorzás adja: 1. Határozzuk meg az y" — y — 2 — 2z1n x differenciálegyenlet általános megoldását (x — 0). ha tudjuk, hogy w(2) — €" és"yalz) — €-? megoldja a hozzá tartozó homogén egye - valr Utejelz), ahol LEZ: a kezdeti feltételből C — 0 és € 6
Megoldás. Az egyenlet másodrendű lineáris, amit elsőrendű rendszerré alakíthatunk y = (y, y′) bevezetésével: y′ = y + 2 x −2x ln x Legyen U(x) = # " ex e−x ex −e−x a megadott megoldásokból álló mátrix (nem szinguláris), ekkor U(x)C a homogén rendszer általános megoldása. Alkalmazzuk az állandók variálásának módszerét, az inhomogén egyenlet megoldása y(x) = c1(x)ex + c2(x)e−x, ahol exc′ 1 c′ 1(x) + e−xc′ 2 c′ 2(x) = 0 exc′ 1 c′ 1(x) −e−xc′ 2 c′ 2(x) = 2 x 2 x −2x ln x. Ebből c′ 1 c′ 1(x) = e−x x x −e−xx ln x c′ 2 c′ 2(x) = −ex x e x + exx ln x. A primitív függvény mindkét esetben a második tag parciális integrálásával kapható meg, például. c1(x) = ! Z e−x x −e−xx ln x dx Z e−x x −e−x(1 + x)1 x Z e−x = e−x(1 + x) ln x + = e−x(1 + x) ln x + dx (−e−x) dx = e−x(1 + x) ln x + e−x + C1, hasonlóan c2(x) = ex(x −1) ln x −ex + C2. Az egyenlet általános megoldása tehát y(x) = 2x ln x + C1ex + C2e−x. 13. Oldjuk meg az xy′′′ + 2y′′ = 1 x diﬀerenciálegyenletet y(1) = 1, y′(1) = y′′(1) = 0 kezdeti feltétellel. Megoldás. A homogén egyenlet y′′ = u-ra nézve elsőrendű: xu′ + 2u = 0, ami szétválasztható: u′ u = −2 x alapján u(x) = C1 x2 . Ezt kétszer integrálva kapjuk a harmadrendű homogén egyenlet általános megoldását: yh(x) = −C1 ln x + C2x + C3. Az eredeti egyenletet átírhatjuk elsőrendű egyenletrendszerré az y = (y, y′, y′′) vektorértékű függvény bevezetésével. A kapott y′ =  y + 1 −2 x  0  1 x2	_Megoldás. Az egyenlet másodrendű lineáris, amit elsőrendű rendszerré alakíthatunk y =_ (y, y[′]) bevezetésével: � 0 � 2 _._ _x_ _[−]_ [2][x][ ln][ x] y[′] = Legyen �0 1� y + 1 0 � _U_ (x) = �e[x] _e[−][x]_ _e[x]_ _−e[−][x]_ a megadott megoldásokból álló mátrix (nem szinguláris), ekkor U (x)C a homogén rendszer általános megoldása. Alkalmazzuk az állandók variálásának módszerét, az inhomogén egyenlet megoldása y(x) = c1(x)e[x] + c2(x)e[−][x], ahol _e[x]c[′]1[(][x][) +][ e][−][x][c][′]2[(][x][) = 0]_ _e[x]c[′]1[(][x][)][ −]_ _[e][−][x][c][′]2[(][x][) = 2]_ _x_ _[−]_ [2][x][ ln][ x.] Ebből _c[′]1[(][x][) =][ e][−][x]_ _x_ _[−]_ _[e][−][x][x][ ln][ x]_ _c[′]2[(][x][) =][ −][e][x]_ _x_ [+][ e][x][x][ ln][ x.] A primitív függvény mindkét esetben a második tag parciális integrálásával kapható meg, például. � [�]e−x � _c1(x) =_ dx _x_ _[−]_ _[e][−][x][x][ ln][ x]_ � [�]e−x = e[−][x](1 + x) ln x + _x_ _[−]_ _[e][−][x][(1 +][ x][)1]x_ � dx � = e[−][x](1 + x) ln x + (−e[−][x]) dx = e[−][x](1 + x) ln x + e[−][x] + C1, hasonlóan _c2(x) = e[x](x −_ 1) ln x − _e[x]_ + C2. Az egyenlet általános megoldása tehát y(x) = 2x ln x + C1e[x] + C2e[−][x]. 13. Oldjuk meg az xy[′′′] + 2y[′′] = [1] _x_ [differenciálegyenletet][ y][(1) = 1,][ y][′][(1) =][ y][′′][(1) = 0 kezdeti] feltétellel. _Megoldás. A homogén egyenlet y[′′]_ = u-ra nézve elsőrendű: xu[′] + 2u = 0, ami szétválasztható: _u[′]_ _u_ [=][ −]x[2] alapján u(x) = _[C][1]_ _x[2][ . Ezt kétszer integrálva kapjuk a harmadrendű homogén egyenlet általá-]_ nos megoldását: yh(x) = −C1 ln x + C2x + C3. Az eredeti egyenletet átírhatjuk elsőrendű egyenletrendszerré az y = (y, y[′], y[′′]) vektorértékű függvény bevezetésével. A kapott   y +  0  01 _x[2]_   y[′] = 0 1 0 0 0 1 0 0 _−_ [2] _x_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet másodrendű lineáris, amit elsőrendű rendszerré alakíthatunk</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> bevezetésével:</span></p> <p style="top:104.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:88.5pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:97.1pt;left:137.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:97.1pt;left:153.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:111.6pt;left:137.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:111.6pt;left:153.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:88.5pt;left:159.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:104.5pt;left:167.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:88.5pt;left:189.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:97.1pt;left:219.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:110.1pt;left:196.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:118.9pt;left:196.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:88.5pt;left:249.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:104.5pt;left:257.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:134.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Legyen</span></p> <p style="top:164.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:149.0pt;left:147.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:157.6pt;left:153.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:157.6pt;left:178.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:172.0pt;left:153.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:172.0pt;left:173.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:149.0pt;left:200.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:195.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a megadott megoldásokból álló mátrix (nem szinguláris), ekkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">C</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a homogén rend-</span></p> <p style="top:210.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">szer általános megoldása. Alkalmazzuk az állandók variálásának módszerét, az inhomogén</span></p> <p style="top:224.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span></p> <p style="top:248.2pt;left:106.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:254.4pt;left:122.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:254.4pt;left:179.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span></sup></p> <p style="top:272.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:278.3pt;left:122.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:278.3pt;left:179.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span></sup></p> <p style="top:280.3pt;left:217.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></sup></p> <p style="top:298.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből</span></p> <p style="top:325.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:332.0pt;left:111.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:334.0pt;left:154.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:353.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:360.0pt;left:111.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:362.1pt;left:160.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></sup></p> <p style="top:380.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A primitív függvény mindkét esetben a második tag parciális integrálásával kapható meg,</span></p> <p style="top:394.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">például.</span></p> <p style="top:425.0pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:412.6pt;left:149.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z </span><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:433.2pt;left:176.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:409.1pt;left:248.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:425.0pt;left:258.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:458.9pt;left:137.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:446.4pt;left:237.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z </span><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:467.1pt;left:264.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)1</span></sup></p> <p style="top:467.1pt;left:346.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:443.0pt;left:354.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:458.9pt;left:364.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:488.9pt;left:137.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:476.4pt;left:237.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:488.9pt;left:247.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:515.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hasonlóan</span></p> <p style="top:538.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1) ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:562.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az egyenlet általános megoldása tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:580.3pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:587.6pt;left:230.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti</span></sup></p> <p style="top:594.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">feltétellel.</span></p> <p style="top:612.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A homogén egyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ra nézve elsőrendű:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xu</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami szétválaszt-</span></p> <p style="top:627.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ható:</span></p> <p style="top:646.2pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:662.4pt;left:109.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:662.4pt;left:144.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:682.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup></p> <p style="top:689.9pt;left:158.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> . Ezt kétszer integrálva kapjuk a harmadrendű homogén egyenlet általá-</span></sup></p> <p style="top:697.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nos megoldását:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">h</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:711.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az eredeti egyenletet átírhatjuk elsőrendű egyenletrendszerré az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vektorértékű</span></p> <p style="top:725.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvény bevezetésével. A kapott</span></p> <p style="top:762.9pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:741.0pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:758.5pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:748.4pt;left:138.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:748.4pt;left:154.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:748.4pt;left:175.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:762.8pt;left:138.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:762.8pt;left:154.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:762.8pt;left:175.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:777.3pt;left:138.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:777.3pt;left:154.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:777.3pt;left:170.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:784.6pt;left:180.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:741.0pt;left:186.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:758.5pt;left:186.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:741.0pt;left:217.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:758.5pt;left:217.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:748.4pt;left:226.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:762.8pt;left:226.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:775.8pt;left:227.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:784.6pt;left:224.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:741.0pt;left:235.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:758.5pt;left:235.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> </div>	page_289.png	.Megoldás. Az egyenlet másodrendű lincáris, amit elsőrendű rendszerré alakíthatunk y — Legyen va-[ a megadott megoldásokból álló mátrix (nem színguláris), ekkor U(r)C a homogén rend- szer általános megoldása. Alkalmazzuk az állandók variálásának módszerét, az inhomog( egyes soldása yíz) — es(rJet 4 eslzje-, ahol édle) t ezéle) - eee A primitív függvény mindkét esetben a második tag parciális integrálásával kapható meg, (.5ztz t feGgds —Ar rjhraet t hasonlóan eldzéle- 1hr ét. Az egyenlet általános megoldása tehát y(z) — 2zlnr.4 CE 4 Cze V) — 1) — 0 kezdeti " 424" — ! dillerenciólegyenletet y(1). -ra nézve elsőrendü: 71 4 2u — 0, ami szétválaszt- .Megoldás. A homogén egyenlet 4" ható: mos megoldását: yi(z) — -C Inz 4 Coz 4 C. függvény bevezetésével, A kapott én egyenlet általá- (y.4. 1) vektorértékü
egyenletrendszer általános megoldása az előbbiek alapján (a korábbi C1 helyett −C1-et írva)   ln x  1 x −1 x2 x2 ·   C1 C2 C3 C1 C2 C3 . Az állandók variálásának módszere szerint az inhomogén egyenlet megoldása y(x) = c1(x)(ln x) + c2(x)x + c3(x), ahol az együtthatófüggvényekre az (ln x)c′ 1 ′ 1(x) + xc′ 2 c′ 2(x) + c′ 3 xc′ 2(x) + c′ 1(x) + c′ 2(x) = 0 3(x) = 0 1 xc′ c′ 1(x) + c′ 2 c′ 2(x) = 0 −1 c′ 1(x) = 1 x2 −1 1 x2 c′ x2 egyenletrendszer teljesül. Ebből c′ 1 c′ 1(x) = −1, c′ 2 c′ 2(x) = 1 x 1 x és c′ 3 l egyenletrendszer teljesül. Ebből c′ 1(x) = −1, c′ 2(x) = 1 x és c′ 3(x) = ln x−1 következik, tehát c1(x) = C1 −x, c2(x) = C2 + ln x és c3(x) = C3 −2x + x ln x. Az általános megoldás és deriváltajai eszerint y(x) = (C1 −x) ln x + (C2 + ln x)x + (C3 −2x + x ln x) = x ln x + C1 ln x + C2x + C3 −2x y′(x) = ln x −1 + C1 C1 x + C2 y′′(x) = 1 x 1 x −C1 x2 C1 x2 . A kezdeti feltételből 1 = y(1) = C2 + C3 −2, 0 = −1 + C1 + C2 = 1 −C1, tehát C1 = 1, C2 = 0 és C3 = 3. A kezdetiérték-probléma megoldása y(x) = 3 −2x + ln x + x ln x. 14. Sorfejtés segítségével határozzuk meg az (1 −x)y′′ + xy′ −y = 0 diﬀerenciálegyenletet y(0) = y′(0) = 1 feltételt kielégítő megoldását. Megoldás. A megoldást X akxk k=0 y(x) = alakban keressük, ennek deriváltjai y′(x) = y′′(x) = X kakxk−1 k=1 ∞ X k(k −1)akxk−2. k=2 Ezeket az egyenletbe helyettesítjük: k(k −1)akxk−2 + x kakxk−1 − X akxk k=0 0 = (1 −x) k=2 k=2 k(k −1)akxk−2 − k=2 k=1 k(k −1)akxk−1 + kakxk − X akxk k=0 ∞ X (k + 2)(k + 1)ak+2xk − k=0 k=1 ∞ X (k + 1)kak+1xk + k=1 kakxk − k=1 X akxk k=0 = 2a2 −a0 + (k + 2)(k + 1)ak+2 −(k + 1)kak+1 + (k −1)ak xk. k=1	egyenletrendszer általános megoldása az előbbiek alapján (a korábbi C1 helyett −C1-et írva) ln x _x_ 1 C1 _x1_ 1 0 _·_ _C2_ _._     _−_ _x[1][2]_ 0 0 _C3_ Az állandók variálásának módszere szerint az inhomogén egyenlet megoldása _y(x) = c1(x)(ln x) + c2(x)x + c3(x),_ ahol az együtthatófüggvényekre az (ln x)c[′]1[(][x][) +][ xc][′]2[(][x][) +][ c][′]3[(][x][) = 0] 1 1[(][x][) +][ c][′]2[(][x][) = 0] _x_ _[c][′]_ _−1_ 1[(][x][) = 1] _x[2][ c][′]_ _x[2]_ egyenletrendszer teljesül. Ebből c[′]1[(][x][) =][ −][1,][ c][′]2[(][x][) =][ 1]x [és][ c]3[′] [(][x][) = ln][ x] _[−]_ [1 következik, tehát] _c1(x) = C1 −_ _x, c2(x) = C2 + ln x és c3(x) = C3 −_ 2x + x ln x. Az általános megoldás és deriváltajai eszerint _y(x) = (C1 −_ _x) ln x + (C2 + ln x)x + (C3 −_ 2x + x ln x) = x ln x + C1 ln x + C2x + C3 − 2x _y[′](x) = ln x −_ 1 + _[C][1]_ _x_ [+][ C][2] _y[′′](x) = x[1]_ _[−]_ _[C]x[2][1][ .]_ A kezdeti feltételből 1 = y(1) = C2 + C3 − 2, 0 = −1 + C1 + C2 = 1 − _C1, tehát C1 = 1,_ _C2 = 0 és C3 = 3. A kezdetiérték-probléma megoldása_ _y(x) = 3 −_ 2x + ln x + x ln x. 14. Sorfejtés segítségével határozzuk meg az (1 − _x)y[′′]_ + xy[′] _−_ _y = 0 differenciálegyenletet_ _y(0) = y[′](0) = 1 feltételt kielégítő megoldását._ _Megoldás. A megoldást_ _y(x) =_ _∞_ � _akx[k]_ _k=0_ alakban keressük, ennek deriváltjai _y[′](x) =_ _y[′′](x) =_ _∞_ � _kakx[k][−][1]_ _k=1_ _∞_ � _k(k −_ 1)akx[k][−][2]. _k=2_ Ezeket az egyenletbe helyettesítjük: _∞_ _∞_ _∞_ 0 = (1 − _x)_ � _k(k −_ 1)akx[k][−][2] + x � _kakx[k][−][1]_ _−_ � _akx[k]_ _k=2_ _k=1_ _k=0_ _∞_ _∞_ _∞_ _∞_ = � _k(k −_ 1)akx[k][−][2] _−_ � _k(k −_ 1)akx[k][−][1] + � _kakx[k]_ _−_ � _akx[k]_ _k=2_ _k=2_ _k=1_ _k=0_ _∞_ _∞_ _∞_ _∞_ = �(k + 2)(k + 1)ak+2x[k] _−_ �(k + 1)kak+1x[k] + � _kakx[k]_ _−_ � _akx[k]_ _k=0_ _k=1_ _k=1_ _k=0_ _∞_ = 2a2 − _a0 +_ � �(k + 2)(k + 1)ak+2 − (k + 1)kak+1 + (k − 1)ak�x[k]. _k=1_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszer általános megoldása az előbbiek alapján (a korábbi</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyett</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-et írva)</span></p> <p style="top:71.7pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:89.3pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:79.1pt;left:114.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:79.1pt;left:143.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:79.1pt;left:160.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:92.0pt;left:121.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:100.9pt;left:121.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:93.6pt;left:144.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:93.6pt;left:160.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:108.0pt;left:113.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:115.3pt;left:123.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:108.0pt;left:144.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:108.0pt;left:160.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:71.7pt;left:166.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:89.3pt;left:166.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:71.7pt;left:181.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:89.3pt;left:181.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:79.1pt;left:188.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:93.6pt;left:188.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:108.0pt;left:188.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:71.7pt;left:201.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:89.3pt;left:201.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:127.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az állandók variálásának módszere szerint az inhomogén egyenlet megoldása</span></p> <p style="top:147.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:168.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol az együtthatófüggvényekre az</span></p> <p style="top:187.7pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:193.8pt;left:141.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xc</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:193.8pt;left:187.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:193.8pt;left:227.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span></sup></p> <p style="top:203.5pt;left:175.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:219.8pt;left:174.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:217.7pt;left:187.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:217.7pt;left:227.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span></sup></p> <p style="top:231.3pt;left:206.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:247.6pt;left:208.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:245.5pt;left:227.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></sup></p> <p style="top:247.6pt;left:265.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:264.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszer teljesül. Ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:270.5pt;left:247.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:270.5pt;left:311.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:271.7pt;left:348.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:270.5pt;left:376.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> következik, tehát</span></sup></p> <p style="top:278.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:293.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az általános megoldás és deriváltajai eszerint</span></p> <p style="top:312.6pt;left:113.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:330.1pt;left:138.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:354.6pt;left:110.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:362.8pt;left:208.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:383.1pt;left:108.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:391.3pt;left:152.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:391.3pt;left:176.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> .</span></i></sup></p> <p style="top:407.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltételből</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:421.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A kezdetiérték-probléma megoldása</span></p> <p style="top:442.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></p> <p style="top:462.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">14. Sorfejtés segítségével határozzuk meg az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span></p> <p style="top:476.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> feltételt kielégítő megoldását.</span></p> <p style="top:493.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megoldást</span></p> <p style="top:520.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:510.6pt;left:147.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:512.0pt;left:144.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:535.7pt;left:144.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:520.3pt;left:161.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:548.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük, ennek deriváltjai</span></p> <p style="top:575.3pt;left:108.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:565.5pt;left:152.7pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:566.9pt;left:149.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:590.7pt;left:149.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:575.3pt;left:166.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:609.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:600.1pt;left:152.7pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:601.5pt;left:149.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:625.2pt;left:149.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2</span></p> <p style="top:609.8pt;left:166.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:638.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ezeket az egyenletbe helyettesítjük:</span></p> <p style="top:664.8pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 = (1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:655.1pt;left:171.7pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:656.5pt;left:168.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:680.2pt;left:168.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2</span></p> <p style="top:664.8pt;left:185.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:655.1pt;left:288.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:656.5pt;left:285.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:680.2pt;left:285.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:664.8pt;left:302.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:655.1pt;left:361.0pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:656.5pt;left:358.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:680.2pt;left:357.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:664.8pt;left:375.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:699.4pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:689.6pt;left:133.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:691.0pt;left:130.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:714.7pt;left:130.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2</span></p> <p style="top:699.4pt;left:147.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:689.6pt;left:242.0pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:691.0pt;left:239.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:714.7pt;left:238.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=2</span></p> <p style="top:699.4pt;left:255.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:689.6pt;left:350.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:691.0pt;left:347.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:714.7pt;left:346.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:699.4pt;left:364.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:689.6pt;left:411.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:691.0pt;left:408.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:714.7pt;left:408.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:699.4pt;left:425.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:733.9pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:724.1pt;left:133.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:725.5pt;left:130.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:749.3pt;left:130.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:733.9pt;left:145.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:724.1pt;left:269.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:725.5pt;left:266.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:749.3pt;left:265.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:733.9pt;left:281.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:724.1pt;left:375.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:725.5pt;left:372.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:749.3pt;left:371.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:733.9pt;left:389.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:724.1pt;left:436.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:725.5pt;left:433.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:749.3pt;left:433.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:733.9pt;left:450.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup></p> <p style="top:768.4pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:758.7pt;left:190.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:760.1pt;left:187.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:783.8pt;left:186.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:758.5pt;left:204.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:768.4pt;left:210.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ka</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:758.5pt;left:444.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:768.4pt;left:450.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> </div>	page_290.png	I-2 0 j la] pdG 40 2 ésejíz) — Inz—1 következik, tehát 22ó e Az általános megoldás és deriváltajai eszerint (C, — 1nr 4 (Cy 4 Inzje 4 (C — 224 2nz) Mr4 Cr Oz 42 20--14 014 C 1-G, tehát C 14. meg az (1— 2)y" 42 — y — 0 differenciálogyenletet ő megoldását. .Megoldás. 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Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 10. feladatsor: Magasabbrendű lineáris diﬀerenciálegyenletek 1. Határozzuk meg az eλx, xeλx, x2eλx, . . . , xk−1eλx függvények Wronski-determinánsát. 2. A Wronski-determináns segítségével határozzuk meg a 4xy′′+2y′+y = 0 diﬀerenciálegyenlet 2. A Wronski-determináns segítségével határozzuk meg a 4xy +2y +y = 0 általános megoldását, ha tudjuk, hogy cos √x megoldja az egyenletet. 3. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását. Határozzuk meg az alábbi diﬀ a) y(4) −2y′′′ −y′′ + 2y′ = 0 b) y′′ −4y′ + 4y = 0 ) y y y c) y′′′ + 3y′′ + 3y′ + y = 0 d) y′′ −4y′ + 29y = 0 ) y y y e) y(4) + 2y′′ + y = 0 4. Legyenek ω ≥0 és α ≥0 valós paraméterek. Oldjuk meg az y′′ + 2αy′ + ω2y = 0 diﬀe- renciálegyenletet y(0) = 1, y′(0) = 0 kezdeti feltétel mellett. Miben különbözik a megoldás α > ω és α < ω esetén? 5. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását. Határozzuk meg az alább a) y′′ −4y′ −12y = xex ) y y y b) y′′ −4y′ −12y = xe−2x ) y y y c) y′′′ −4y′′ + 4y′ = x2 + e2x ) y y y d) y′′ −2y′ + 5y = ex sin 2x További gyakorló feladatok 6. Bizonyítsuk be, hogy az y′ 1 y′ 1 = y2 y′ 2 = −e y′ 2 = −e2xy1 + y2 diﬀerenciálegyenlet-rendszernek létezik nem korlátos megoldása. 7. Határozzuk meg az y′′ −y′ −e2xy = 0 diﬀerenciálegyenlet általános megoldását, ha tudjuk, 7. Határozzuk meg az y y e y = 0 diﬀerenciálegyenlet általános meg hogy eex megoldás. 8. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását. g a) y′′ + 2y′ + 10y = 0 b) y′′ −12y′ + 27y = 0 ) y y y c) y′′ −10y′ + 25y = 0 d) y(4) + 18y′′ + 81y = 0 ) y y y e) y′′′ −6y′′ + 12y′ −8y = 0 ) y y y y f) y(n) −y = 0, ahol n ≥1 egész 9. Legyenek ω0 > 0 és ω ≥0 valós paraméterek. Oldjuk meg az y′′ + ω2 0y 9. Legyenek ω0 > 0 és ω ≥0 valós paraméterek. Oldjuk meg az y′′ + ω2 0y = sin(ωx) diﬀerenciálegyenletet y(0) = 0, y′(0) = 0 kezdeti feltétel mellett. Mi történik, ha ω = ω0? 10. Legyenek ω1, ω2 ≥0 valós paraméterek. Oldjuk meg az y(4) + (ω2 2)y′′ + ω2 1ω2 2y = 0 1 + ω2 ω2 2 1 + ω2 ω2 2)y′′ + ω2 1 ω2 1ω2 2 10. Legyenek ω1, ω2 ≥0 valós paraméterek. Oldjuk meg az y(4) + (ω2 2)y′′ + ω2 1ω2 1 + ω2 2y = 0 diﬀerenciálegyenletet y(0) = 1, y′(0) = y′′(0) = y′′′(0) = 0 kezdeti feltétel mellett. Mi történik, ha ω1 = ω2? 11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását. g p a) y′′ + 4y′ + 8y = e−2x cos 2x, y(0) = 1, y′(0) = 0	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 10. feladatsor: Magasabbrendű lineáris differenciálegyenletek 1. Határozzuk meg az e[λx], xe[λx], x[2]e[λx], . . ., x[k][−][1]e[λx] függvények Wronski-determinánsát. 2. A Wronski-determináns segítségével határozzuk meg a 4xy[′′]+2y[′]+y = 0 differenciálegyenlet általános megoldását, ha tudjuk, hogy cos _[√]x megoldja az egyenletet._ 3. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) y[(4)] _−_ 2y[′′′] _−_ _y[′′]_ + 2y[′] = 0 b) y[′′] _−_ 4y[′] + 4y = 0 c) y[′′′] + 3y[′′] + 3y[′] + y = 0 d) y[′′] _−_ 4y[′] + 29y = 0 e) y[(4)] + 2y[′′] + y = 0 4. Legyenek ω ≥ 0 és α ≥ 0 valós paraméterek. Oldjuk meg az y[′′] + 2αy[′] + ω[2]y = 0 differenciálegyenletet y(0) = 1, y[′](0) = 0 kezdeti feltétel mellett. Miben különbözik a megoldás _α > ω és α < ω esetén?_ 5. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) y[′′] _−_ 4y[′] _−_ 12y = xe[x] b) y[′′] _−_ 4y[′] _−_ 12y = xe[−][2][x] c) y[′′′] _−_ 4y[′′] + 4y[′] = x[2] + e[2][x] d) y[′′] _−_ 2y[′] + 5y = e[x] sin 2x ## További gyakorló feladatok 6. Bizonyítsuk be, hogy az _y1[′]_ [=][ y][2] _y2[′]_ [=][ −][e][2][x][y][1] [+][ y][2] differenciálegyenlet-rendszernek létezik nem korlátos megoldása. 7. Határozzuk meg az y[′′] _−_ _y[′]_ _−_ _e[2][x]y = 0 differenciálegyenlet általános megoldását, ha tudjuk,_ hogy e[e][x] megoldás. 8. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) y[′′] + 2y[′] + 10y = 0 b) y[′′] _−_ 12y[′] + 27y = 0 c) y[′′] _−_ 10y[′] + 25y = 0 d) y[(4)] + 18y[′′] + 81y = 0 e) y[′′′] _−_ 6y[′′] + 12y[′] _−_ 8y = 0 f) y[(][n][)] _−_ _y = 0, ahol n ≥_ 1 egész 9. Legyenek ω0 > 0 és ω ≥ 0 valós paraméterek. Oldjuk meg az y[′′] + ω0[2][y][ = sin(][ωx][) differen-] ciálegyenletet y(0) = 0, y[′](0) = 0 kezdeti feltétel mellett. Mi történik, ha ω = ω0? 10. Legyenek ω1, ω2 ≥ 0 valós paraméterek. Oldjuk meg az y[(4)] + (ω1[2] [+][ ω]2[2][)][y][′′][ +][ ω]1[2][ω]2[2][y][ = 0] differenciálegyenletet y(0) = 1, y[′](0) = y[′′](0) = y[′′′](0) = 0 kezdeti feltétel mellett. Mi történik, ha ω1 = ω2? 11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását. a) y[′′] + 4y[′] + 8y = e[−][2][x] cos 2x, y(0) = 1, y[′](0) = 0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">10. feladatsor: Magasabbrendű lineáris</span></b></p> <p style="top:106.3pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">diﬀerenciálegyenletek</span></b></p> <p style="top:146.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvények Wronski-determinánsát.</span></p> <p style="top:162.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. A Wronski-determináns segítségével határozzuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span></p> <p style="top:177.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldását, ha tudjuk, hogy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldja az egyenletet.</span></p> <p style="top:193.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását.</span></p> <p style="top:208.2pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></p> <p style="top:224.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:241.0pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:257.5pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 29</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:273.9pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:290.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀe-</span></p> <p style="top:304.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">renciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Miben különbözik a megoldás</span></p> <p style="top:319.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α > ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α < ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén?</span></p> <p style="top:335.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását.</span></p> <p style="top:350.1pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:366.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:383.0pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:399.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:432.1pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:456.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Bizonyítsuk be, hogy az</span></p> <p style="top:481.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:487.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:499.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:505.2pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:524.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszernek létezik nem korlátos megoldása.</span></p> <p style="top:541.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását, ha tudjuk,</span></p> <p style="top:555.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás.</span></p> <p style="top:571.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását.</span></p> <p style="top:586.3pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:602.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 27</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:619.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 25</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:635.7pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 18</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 81</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:652.1pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:668.5pt;left:83.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">f)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egész</span></p> <p style="top:685.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:691.1pt;left:427.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀeren-</span></sup></p> <p style="top:699.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Mi történik, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:715.9pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:722.0pt;left:418.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:722.0pt;left:446.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:722.0pt;left:490.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:722.0pt;left:502.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup></p> <p style="top:730.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Mi</span></p> <p style="top:744.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">történik, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:761.2pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását.</span></p> <p style="top:775.6pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span></p> </div>	page_292.png	Matematika A3 gyakorlat Energetika és Mechatzonika TsSc szakok, 2016/17 ősz 10. feladatsor: Magasabbrendű lincáris differenciálegyenletek 1. Határozzuk meg az €, rel", ter 291 függvények Wronski-determinánsát A Wronski-determináns segítsé 1 határozzuk meg a 4z/-4-24/4-y — ( differenciálegye a) 92" 2/ -0 b) — dy 449 C AKATB a) - 4y4-294—0 e 2Y é 4. Legyenek a 2 0 és a 2 0 valós paraméterek. Oldjuk meg az y" -- 2a9 4. renciálegyendetet y(0). Py — 0 dílfe- tel mellett. Miben különbözik a megoldás a) 4 -lr 1) — 4y— 2eee OY éot ét d) 2Y 4őy — €sin2r "További gyakorló feladatok 6. Bizonyítsuk be, hogy az M FEE dilferenciálegyet dszernek létezik nem korlátos megoldása. DEZEZTEN 1) — 12 4274 e) /— 10 4254 a) 94 18474-8ly—0 ely" 64 12/-84—0 949 9. Legyenek aa 2. 0 és 4 2 0 valós paraméterek. Oldjuk meg az y" 4-ugy — sin(2oz) dífferes csálegyenletet y(0) — 0, (0) — 0 kezdeti feltétel mellett. Mi történik, ha 10. Legyenek az,s4 2 0 valós paraméterek. Oldjuk meg az y09 4. (s2 4 s2ly" differenciálegyenletet y(0) — 1, y(0) — y"(0) — y"(0) — 0 kezdeti felt történik, ha ax — ? 0 differenciálegyenlet általános megoldását, ha tudjuk, 0. ahol n 2 1 egész 11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását. a) 41 4-4y-4.8y — €-2 c0s2r, y(0) — 1. Y(0)—0
12. Legyenek ω0, ω, α > 0 valós paraméterek. Keressük meg az y′′ + 2αy′ + ω2 0y = sin(ωx) differenciálegyenletet periodikus megoldását (y(x) = C cos(ωx) + D sin(ωx) alakban). Milyen ω mellett maximális y illetve y′ amplitúdója? Megoldás. A homogén egyenlet karakterisztikus polinomjának gyökei λ± = −α± q α2 −ω2 0 q α2 −ω2 Megoldás. A homogén egyenlet karakterisztikus polinomjának gyökei λ± = −α± 0, ezeknek mindig negatív a valós része, tehát a homogén egyenletnek nincs nemtriviális periodikus megoldása. Nincsen külső rezonancia, mert az inhomogén tagban a ±iωx kitevők tisztán képzetesek. A periodikus megoldás ezek szerint csak y(x) = C cos(ωx) + D sin(ωx) alakú lehet. y′(x) = Dω cos(ωx) −Cω sin(ωx) y′′(x) = −Cω2 cos(ωx) −Dω2 sin(ωx) felhasználásával behelyettesítés után az egyenlet (ω2 0 ω2 0C + 2αωD −ω2C) cos(ωx) + (ω2 0 ω2 0D −2αωC −ω2D) cos(ωx) = sin(ωx) lesz. Ebből C = − 2αω (ω2 −ω2 0)2 + ω2 0)2 + 4α2ω2 ( 0) D = − ω2 −ω2 0 ( 2 2)2 + 4 (ω2 −ω2 0 ω0 ω2 0)2 + 4α2ω2. Érdemes a kapott megoldást A sin(ωx −ϕ) alakba is átírni, ekkor A a rezgés amplitúdója, ϕ pedig az inhomogén taghoz képest mért fázis. Az addíciós képlet felhasználásával A = p g g √ C2 + D2 és ϕ = arccos C √ C2+D2, tehát A = q (ω2 −ω2 0)2 + 4α2ω2 q (ω2 −ω2 0 ϕ = arccos ω2 0 q ( 2 2 0 −ω2 q (ω2 −ω2 0 ω ω2 0)2 + 4α2ω2. q Függvényvizsgálattal meggyőződhetünk róla, hogy adott ω0, α mellett az amplitúdó akkor maximális, ha ω = max{0, ω2 0 −2α2}. q max{0, ω2 0 q maximális, ha ω = max{0, ω2 0 −2α2}. y′ = Aω cos(ωx −ϕ) amplitúdója Aω, ez ω = ω0 esetén maximális.	12. Legyenek ω0, ω, α > 0 valós paraméterek. Keressük meg az y[′′] + 2αy[′] + ω0[2][y][ = sin(][ωx][) dif-] ferenciálegyenletet periodikus megoldását (y(x) = C cos(ωx) + D sin(ωx) alakban). Milyen _ω mellett maximális y illetve y[′]_ amplitúdója? � _Megoldás. A homogén egyenlet karakterisztikus polinomjának gyökei λ± = −α±_ _α[2]_ _−_ _ω0[2][,]_ ezeknek mindig negatív a valós része, tehát a homogén egyenletnek nincs nemtriviális periodikus megoldása. Nincsen külső rezonancia, mert az inhomogén tagban a ±iωx kitevők tisztán képzetesek. A periodikus megoldás ezek szerint csak y(x) = C cos(ωx) + D sin(ωx) alakú lehet. _y[′](x) = Dω cos(ωx) −_ _Cω sin(ωx)_ _y[′′](x) = −Cω[2]_ cos(ωx) − _Dω[2]_ sin(ωx) felhasználásával behelyettesítés után az egyenlet (ω0[2][C][ + 2][αωD][ −] _[ω][2][C][) cos(][ωx][) + (][ω]0[2][D][ −]_ [2][αωC][ −] _[ω][2][D][) cos(][ωx][) = sin(][ωx][)]_ lesz. Ebből 2αω _C = −_ (ω[2] _−_ _ω0[2][)][2][ + 4][α][2][ω][2]_ _D = −_ _ω[2]_ _−_ _ω0[2]_ (ω[2] _−_ _ω0[2][)][2][ + 4][α][2][ω][2]_ _[.]_ Érdemes a kapott megoldást A sin(ωx − _ϕ) alakba is átírni, ekkor A a rezgés amplitúdója,_ _ϕ pedig az inhomogén taghoz képest mért fázis. Az addíciós képlet felhasználásával A =_ _√_ _C_ [2] + D[2] és ϕ = arccos _√_ _C_ _C[2]+D[2]_ [, tehát] 1 _A =_ � (ω[2] _−_ _ω0[2][)][2][ + 4][α][2][ω][2]_ _ω0[2]_ _[−]_ _[ω][2]_ _ϕ = arccos_ � (ω[2] _−_ _ω0[2][)][2][ + 4][α][2][ω][2]_ _[.]_ Függvényvizsgálattal meggyőződhetünk róla, hogy adott ω0, α mellett az amplitúdó akkor � maximális, ha ω = max{0, ω0[2] _[−]_ [2][α][2][}][.] _y[′]_ = Aω cos(ωx − _ϕ) amplitúdója Aω, ez ω = ω0 esetén maximális._ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, ω, α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Keressük meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:65.2pt;left:452.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> dif-</span></sup></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ferenciálegyenletet periodikus megoldását (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban). Milyen</span></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mellett maximális</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> illetve</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amplitúdója?</span></p> <p style="top:109.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A homogén egyenlet karakterisztikus polinomjának gyökei</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i></p> <p style="top:98.7pt;left:486.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:109.4pt;left:496.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:115.2pt;left:530.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:123.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezeknek mindig negatív a valós része, tehát a homogén egyenletnek nincs nemtriviális pe-</span></p> <p style="top:138.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">riodikus megoldása. Nincsen külső rezonancia, mert az inhomogén tagban a</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">iωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kitevők</span></p> <p style="top:152.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tisztán képzetesek. A periodikus megoldás ezek szerint csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:167.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú lehet.</span></p> <p style="top:193.6pt;left:108.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Dω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Cω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:211.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Cω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Dω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:237.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felhasználásával behelyettesítés után az egyenlet</span></p> <p style="top:263.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:270.0pt;left:118.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αωD</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:270.0pt;left:286.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αωC</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:290.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lesz. Ebből</span></p> <p style="top:318.8pt;left:107.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:310.7pt;left:181.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αω</span></i></p> <p style="top:327.0pt;left:142.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:332.8pt;left:181.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:351.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:343.5pt;left:172.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:349.6pt;left:206.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:359.7pt;left:142.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:365.5pt;left:181.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:386.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Érdemes a kapott megoldást</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakba is átírni, ekkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a rezgés amplitúdója,</span></p> <p style="top:401.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pedig az inhomogén taghoz képest mért fázis. Az addíciós képlet felhasználásával</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:405.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:415.5pt;left:87.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = arccos</span></p> <p style="top:414.0pt;left:220.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">C</span></i></p> <p style="top:417.0pt;left:205.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:423.7pt;left:212.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">C</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">D</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></sup></p> <p style="top:450.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:442.0pt;left:183.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:450.8pt;left:132.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:461.4pt;left:142.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:467.2pt;left:181.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:488.3pt;left:107.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = arccos</span></p> <p style="top:480.2pt;left:199.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:486.4pt;left:207.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:489.1pt;left:165.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:499.7pt;left:175.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:505.5pt;left:214.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:526.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Függvényvizsgálattal meggyőződhetünk róla, hogy adott</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mellett az amplitúdó akkor</span></p> <p style="top:542.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">maximális, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:532.0pt;left:175.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:542.5pt;left:185.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">max</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">{</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:548.3pt;left:232.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">}</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:557.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Aω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ϕ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> amplitúdója</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Aω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ez</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén maximális.</span></p> <p style="top:805.5pt;left:291.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span></p> </div>	page_293.png	12. Legyenek say.4.a — 0 valós paraméterek. Keressük meg az y 4 2aY 4-sly ferenciálegyenletet periodikus megoldását (y(z) — C cosloz) 4 Dsin(-or) alakban). Milyen . Megoldás. A homogén egyenlet karakterisztikus polinomjának győkei A, — ezeknek mindig negatív a valós része, tehát a homogén egyenlet riodikus megoldása. Nincsen külső rezonancia, mert az inhomogén tagban a íor kitevők tisztán képzetesek. A periodikus megoldás ezek szerint csak y(r) — Ccos(tor) 4 Dsin(or) alakú lehet. ví) — Ducos(uz) 1( — C c0sl felhasználásával behelyettesítés után az egyenlet ( lesz. Ebből C 4 2aD — PCY coslcsz) 4. D — 20100 — s? D) cosi c- podig az iahomogén taghoz képest mést fázis. Az addíciós képlet felhasználásával A Z ÉT DA és . — azecos gzőrra, tehát . 1 MEE Függyényvizsgálattal meggyőződhetünk róla, hogy adott ax. a mellett az amplitádó akkor A — 4nas(0, ss — 2a). maximmális, ha 4 — y/max(0,j — 202). W 2 Acoslor — ) amplitúdója Azz, ez 4 — c esetén maximális. 19
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 10. feladatsor: Magasabbrendű lineáris diﬀerenciálegyenletek (megoldás) 1. Határozzuk meg az eλx, xeλx, x2eλx, . . . , xk−1eλx függvények Wronski-determinánsát. Megoldás. A megadott függvények eλxf(x) alakúak, ezek deriváltjait a Leibniz-szabály alkalmazásával állíthatjuk elő: dn dxneλxf(x) = i=0 λif (n−i)(x)eλx = f (n)(x)eλx + ! n X λif (n−i)(x)eλx. i i=1 Ebben az alakban láthatjuk, hogy az összegben megjelenő tagok mindegyike előáll eλxf(x) legfeljebb n −1-edik deriváltjainak lineáris kombinációjaként, ahol az együtthatók f-től nem függnek. Így például eλxf1(x) eλxf2(x) (eλxf1(x))′ (eλxf2(x))′ eλxf1(x) eλxf2(x) λeλxf1(x) + eλxf ′ 1(x) λeλxf2(x) + eλx f ′ 1(x) λeλxf2(x) + eλxf ′ 2 f ′ 2(x) eλxf1(x) eλxf2(x) eλxf ′ 1(x) eλxf ′ 2(x) f ′ 1(x) eλxf ′ 2 f ′ 2(x) = e2λx f1(x) f2(x) f ′ 1(x) f ′ 2(x) f ′ 1(x) f ′ 2 f ′ 2(x) Hasonlóan általában is elérhető sorműveletekkel, hogy csak f (j) i (x) elemek maradjanak a determinánsban: Hasonlóan általában is elérhető sorműveletekkel, hogy csak f (j) i f1(x) f2(x) · · · fk(x) f ′ k(x) 1(x) f ′ 2(x) · · · f ′ eλxf1(x) · · · eλxfk(x) ... ... ... dk−1 dxk−1eλxf1(x) · · · dk−1 dxk−1eλxfk(x) f ′ 1(x) f ′ 2 f ′ k 2(x) · · · f ′ f ′ 1(x) f ′ 2(x) · · · f ′ k(x) ... ... ... ... f (k−1) 1 (x) f (k−1) 2 (x) · · · f (k−1) k (x = ekλx (x) f (k−1) 2 (x) · · · f (k−1) k (x) A megadott függvények Wronski-determinánsa tehát 1 x x2 · · · xk−1 0 1 2x · · · (k −1)xk−2 0 0 2 · · · (k −1)(k −2)xk−3 ... ... ... ... ... 0 0 0 · · · (k −1)! k−1 Y n!. n=0 W(x) = ekλx = ekλx Azt is észrevehetjük, hogy a megadott függvények az !k d dx −λ y(x) = 0, azaz ! k k X (−λ)k−ny(n) = 0 n n=0	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 10. feladatsor: Magasabbrendű lineáris differenciálegyenletek (megoldás) 1. Határozzuk meg az e[λx], xe[λx], x[2]e[λx], . . ., x[k][−][1]e[λx] függvények Wronski-determinánsát. _Megoldás. A megadott függvények e[λx]f_ (x) alakúak, ezek deriváltjait a Leibniz-szabály alkalmazásával állíthatjuk elő: d[n] dx[n] _[e][λx][f]_ [(][x][) =] _n_ � _i=0_ �n� _λ[i]f_ [(][n][−][i][)](x)e[λx] = f [(][n][)](x)e[λx] + _i_ _n_ � _i=1_ �n� _λ[i]f_ [(][n][−][i][)](x)e[λx]. _i_ Ebben az alakban láthatjuk, hogy az összegben megjelenő tagok mindegyike előáll e[λx]f (x) legfeljebb n − 1-edik deriváltjainak lineáris kombinációjaként, ahol az együtthatók f -től nem függnek. Így például _e[λx]f1(x)_ _e[λx]f2(x)_ (e[λx]f1(x))[′] (e[λx]f2(x))[′] �� = _e[λx]f1(x)_ _e[λx]f2(x)_ _λe[λx]f1(x) + e[λx]f1[′][(][x][)]_ _λe[λx]f2(x) + e[λx]f2[′][(][x][)]_ �� = _e[λx]f1(x)_ _e[λx]f2(x)_ _e[λx]f1[′][(][x][)]_ _e[λx]f2[′][(][x][)]_ �� = e[2][λx] _f1(x)_ _f2(x)_ _._ _f1[′][(][x][)]_ _f2[′][(][x][)]_ �� Hasonlóan általában is elérhető sorműveletekkel, hogy csak fi[(][j][)][(][x][) elemek maradjanak a] determinánsban: _f1(x)_ _f2(x)_ _· · ·_ _fk(x)_ _f1[′][(][x][)]_ _f2[′][(][x][)]_ _· · ·_ _fk[′]_ [(][x][)] ... ... ... ... _f1[(][k][−][1)](x)_ _f2[(][k][−][1)](x)_ _· · ·_ _fk[(][k][−][1)](x)��_ �� _e[λx]f1(x)_ _· · ·_ _e[λx]fk(x)_ ... ... ... = e[kλx] d[k][−][1] _· · ·_ d[k][−][1] dx[k][−][1] _[e][λx][f][1][(][x][)]_ dx[k][−][1] _[e][λx][f][k][(][x][)]��_ _._ A megadott függvények Wronski-determinánsa tehát �� _W_ (x) = e[kλx] 1 _x_ _x[2]_ _· · ·_ _x[k][−][1]_ 0 1 2x _· · ·_ (k − 1)x[k][−][2] 0 0 2 _· · ·_ (k − 1)(k − 2)x[k][−][3] ... ... ... ... ... 0 0 0 _· · ·_ (k − 1)! �� _k−1_ = e[kλx] � _n!._ _n=0_ Azt is észrevehetjük, hogy a megadott függvények az � d dx _[−]_ _[λ]_ �k _y(x) = 0,_ azaz _k_ � _n=0_ �k� (−λ)[k][−][n]y[(][n][)] = 0 _n_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">10. feladatsor: Magasabbrendű lineáris</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">diﬀerenciálegyenletek (megoldás)</span></b></p> <p style="top:149.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvények Wronski-determinánsát.</span></p> <p style="top:168.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A megadott függvények</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakúak, ezek deriváltjait a Leibniz-szabály al-</span></p> <p style="top:183.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kalmazásával állíthatjuk elő:</span></p> <p style="top:208.5pt;left:111.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup></p> <p style="top:224.8pt;left:107.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup></p> <p style="top:206.8pt;left:186.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></p> <p style="top:208.2pt;left:181.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:231.7pt;left:182.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:200.7pt;left:198.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i></p> <p style="top:224.8pt;left:207.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i></p> <p style="top:200.7pt;left:213.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:216.6pt;left:221.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:206.8pt;left:376.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></p> <p style="top:208.2pt;left:372.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:231.7pt;left:372.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:200.7pt;left:388.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i></p> <p style="top:224.8pt;left:397.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i></p> <p style="top:200.7pt;left:403.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:216.6pt;left:411.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:252.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebben az alakban láthatjuk, hogy az összegben megjelenő tagok mindegyike előáll</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:266.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">legfeljebb</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-edik deriváltjainak lineáris kombinációjaként, ahol az együtthatók</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-től</span></p> <p style="top:281.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nem függnek. Így például</span></p> <p style="top:298.1pt;left:108.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:307.1pt;left:117.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:307.1pt;left:181.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:321.5pt;left:111.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:321.5pt;left:175.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:298.1pt;left:229.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:298.1pt;left:248.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:307.1pt;left:283.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:307.1pt;left:398.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:321.5pt;left:251.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:327.7pt;left:336.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:321.5pt;left:366.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:327.7pt;left:451.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:298.1pt;left:471.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:348.3pt;left:235.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:332.0pt;left:248.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:341.0pt;left:251.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:341.0pt;left:303.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:355.4pt;left:251.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:361.6pt;left:273.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:355.4pt;left:303.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:361.6pt;left:324.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:332.0pt;left:345.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:382.2pt;left:235.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup></p> <p style="top:365.8pt;left:270.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:374.8pt;left:273.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:374.8pt;left:309.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:389.3pt;left:273.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:395.4pt;left:279.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:389.3pt;left:309.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:395.4pt;left:315.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:365.8pt;left:336.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> .</span></i></p> <p style="top:419.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hasonlóan általában is elérhető sorműveletekkel, hogy csak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">j</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:425.1pt;left:395.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> elemek maradjanak a</span></sup></p> <p style="top:433.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">determinánsban:</span></p> <p style="top:456.6pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:464.1pt;left:122.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:464.1pt;left:187.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:464.1pt;left:224.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:475.6pt;left:142.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:476.6pt;left:188.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:475.6pt;left:243.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:498.0pt;left:113.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">d</span><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup></p> <p style="top:506.9pt;left:111.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:499.6pt;left:187.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:498.0pt;left:214.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">d</span><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup></p> <p style="top:506.9pt;left:212.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:456.6pt;left:279.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:481.9pt;left:286.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kλx</span></i></sup></p> <p style="top:450.6pt;left:320.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:456.7pt;left:333.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:456.7pt;left:389.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:456.7pt;left:434.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:456.7pt;left:468.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:471.1pt;left:333.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:477.3pt;left:339.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:471.1pt;left:389.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:477.3pt;left:394.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:471.1pt;left:434.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:471.1pt;left:468.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:477.3pt;left:473.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:482.6pt;left:345.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:482.6pt;left:400.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:483.6pt;left:435.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:482.6pt;left:479.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:507.0pt;left:324.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup></p> <p style="top:512.8pt;left:329.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:507.0pt;left:353.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:507.0pt;left:379.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup></p> <p style="top:512.8pt;left:385.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:507.0pt;left:409.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:507.0pt;left:434.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:507.0pt;left:458.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup></p> <p style="top:513.2pt;left:464.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:507.0pt;left:488.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:450.6pt;left:504.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:481.9pt;left:509.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:532.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A megadott függvények Wronski-determinánsa tehát</span></p> <p style="top:589.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kλx</span></i></sup></p> <p style="top:549.4pt;left:172.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:558.1pt;left:176.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:558.1pt;left:192.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:558.1pt;left:209.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:558.1pt;left:231.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:558.1pt;left:291.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:572.6pt;left:176.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:572.6pt;left:192.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:572.6pt;left:208.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:572.6pt;left:231.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:572.6pt;left:273.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:587.0pt;left:176.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:587.0pt;left:192.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:587.0pt;left:212.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:587.0pt;left:231.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:587.0pt;left:255.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:598.5pt;left:177.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:598.5pt;left:193.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:598.5pt;left:213.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:599.5pt;left:231.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:598.5pt;left:300.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:620.9pt;left:176.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:620.9pt;left:192.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:620.9pt;left:212.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:620.9pt;left:231.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:620.9pt;left:282.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)!</span></p> <p style="top:549.4pt;left:349.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:589.6pt;left:356.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kλx</span></i></sup></p> <p style="top:579.9pt;left:391.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:581.3pt;left:392.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Y</span></p> <p style="top:604.6pt;left:391.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:589.6pt;left:409.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">!</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:646.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Azt is észrevehetjük, hogy a megadott függvények az</span></p> <p style="top:666.3pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></p> <p style="top:690.5pt;left:115.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup></p> <p style="top:666.3pt;left:151.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:682.3pt;left:166.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:712.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span></p> <p style="top:735.0pt;left:112.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:736.4pt;left:107.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:759.7pt;left:106.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:728.8pt;left:124.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i></p> <p style="top:753.0pt;left:132.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i></p> <p style="top:728.8pt;left:139.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:744.8pt;left:147.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></p> </div>	page_294.png	Matematika A3 gyakorlat Energetika és Mechatzonika TsSc szakok, 2016/17 ősz 10. feladatsor: Magasabbrendű lincáris differenciálegyenletek (megoldás) 1. Határozzuk meg az €, eV 27é.. . , 1412 függyények Wronski-determinánsát "Megoldás. A megadott függvények e" /(z) alakúak, ezek deriváltjait a Leibniz-szabály al- kalmazásával állíthatjuk elő: éo :É(;')A'/"' s ár n K9 ()A oo Ebben az alakban láthatjuk, hogy az összegben megjelenő tagok mindegyike előáll e" /(r) tegfeljebb n — 1-edik deriváltjainak lincáris kombinációjaként, ahol az együtthatók f-től sek. Így mem £ ául Arh eh [ (eshizy (ezfányi l erhíz rfele) bezhg téh xezflej t szhln) eh ézple) erfia) ér hlő) 1o fiíe) lo determinánsban: Hason Az z h ASI hár) én ASL YO Y S] fiz) eeeh - £h y AG . 1E A megadott fűggvények Wronski-determinánsa tehát 12 Fa 0 1 2- (Foyet j Ww(e)-ebef0 0 2 .. (E-1(R-9 - T e. poc (F Azt is észrevehetjük, hogy a megadott függyények az
lineáris diﬀerenciálegyenlet megoldásterének bázisát alkotját. Elsőrendű egyenletrendszerré alakítva y = (y, y′, . . . , y(k−1)) bevezetésével az egyenlet · · · 0 0 1 ... ... ... ... ... ... 0 0 0 0 · · · 1   (−λ)  y′ = y = Ay k (−λ)k−1 k (−λ)k−2 k (−λ)k−3 · · · k k−1 lesz, tehát a Wronski-determinánsra a W ′ = (Tr A)W = kλW diﬀerenciálegyenlet teljesül, emiatt W(x) = ekλxW(0). 2. A Wronski-determináns segítségével határozzuk meg a 4xy′′+2y′+y = 0 diﬀerenciálegyenlet általános megoldását, ha tudjuk, hogy cos √x megoldja az egyenletet. Megoldás. Az a2(x)y′′ + a1(x)y′ + a0(x)y = 0 diﬀerenciálegyenlet ekvivalens az y′ = −a0(x) a2(x) a0(x) a2(x) −a1(x) a2(x) a2(x) elsőrendű egyenletrendszerrel, ennek Wronski-determinánsa teljesíti a W ′ = −a1(x) a2(x)W egyenletet. Most a1(x) = 2 a2(x) = 4x, tehát W ′ = −1 2xW, amiből W(x) = C √x. Másrészt ha y1 és y2 az eredeti egyenlet két megoldása, akkor a W(x) = y1(x) y2(x) y′ 1(x) y′ 2(x) y′ 1(x) y′ 2 y′ 2(x) függvény ugyanezt a diﬀerenciálegyenletet teljesíti. Ha y1(x) = cos √x ismert, akkor ez y2-re nézve elsőrendű lineáris inhomogén diﬀerenciálegyenletet jelent, amit az állandók variálásának módszerével oldhatunk meg. A konstrukció miatt y1 megoldja a homogén egyenletet, tehát az inhomogén egyenlet megoldását c(x)y1(x) alakban keressük. y1(x) c(x)y1(x) y′ 1(x) (c(x)y1(x))′ y1(x) c(x)y1(x) y′ 1(x) (c(x)y1(x)) y1(x) c(x)y1(x) y′ 1(x) c(x)y′ 1(x) + c′(x y′ 1(x) c(x)y′ 1 y′ 1(x) + c′(x)y1(x) y1(x) y′ 1(x) y′ 1(x) c′(x)y1(x) = c′(x)y1(x)2 felhasználásával és pl. C = 1 választásával most c′(x) = 1 cos2 √x 1 √x, tehát c(x) = 2 tan √x. Így y2(x) = 2 sin √x, és az általános megoldás A cos √x + B sin √x. 3. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását. a) y(4) −2y′′′ −y′′ + 2y′ = 0 b) y′′ −4y′ + 4y = 0 c) y′′′ + 3y′′ + 3y′ + y = 0 d) y′′ −4y′ + 29y = 0	lineáris differenciálegyenlet megoldásterének bázisát alkotját. Elsőrendű egyenletrendszerré alakítva y = (y, y[′], . . ., y[(][k][−][1)]) bevezetésével az egyenlet y[′] =  0 1 0 _· · ·_ 0  0 0 1 ... ... ... ... ... ... 0 0 0 0 _· · ·_ 1 −�k�(−λ)[k][−][1] _−�k�(−λ)[k][−][2]_ _−�k�(−λ)[k][−][3]_ _· · ·_ _−�_ _k_ �(−λ) 1 2 3 _k−1_ y = Ay lesz, tehát a Wronski-determinánsra a W _[′]_ = (Tr A)W = kλW differenciálegyenlet teljesül, emiatt W (x) = e[kλx]W (0). 2. A Wronski-determináns segítségével határozzuk meg a 4xy[′′]+2y[′]+y = 0 differenciálegyenlet általános megoldását, ha tudjuk, hogy cos _[√]x megoldja az egyenletet._ _Megoldás. Az a2(x)y[′′]_ + a1(x)y[′] + a0(x)y = 0 differenciálegyenlet ekvivalens az y[′] = � 0 1 _−_ _[a][0][(][x][)]_ _−_ _[a][1][(][x][)]_ _a2(x)_ _a2(x)_ � y elsőrendű egyenletrendszerrel, ennek Wronski-determinánsa teljesíti a _W_ _[′]_ = −[a][1][(][x][)] _a2(x)[W]_ egyenletet. Most a1(x) = 2 a2(x) = 4x, tehát W _[′]_ = − 2[1]x _[W]_ [, amiből][ W] [(][x][) =] _√Cx_ . Másrészt ha y1 és y2 az eredeti egyenlet két megoldása, akkor a _W_ (x) = _y1(x)_ _y2(x)_ _y1[′]_ [(][x][)] _y2[′]_ [(][x][)] �� függvény ugyanezt a differenciálegyenletet teljesíti. Ha y1(x) = cos _[√]x ismert, akkor ez_ _y2-re nézve elsőrendű lineáris inhomogén differenciálegyenletet jelent, amit az állandók_ variálásának módszerével oldhatunk meg. A konstrukció miatt y1 megoldja a homogén egyenletet, tehát az inhomogén egyenlet megoldását c(x)y1(x) alakban keressük. _yy11[′]_ [(]([x]x[)]) (cc((xx))yy11((xx))) _[′]_ = _yy11[′]_ ([(]x[x])[)] _c(x)y1[′]_ [(]c[x](x[) +])y1[ c]([′]x[(][x]) [)][y][1][(][x][)] = _yy11[′]_ ([(]x[x])[)] _c[′](x)0y1(x)_ = c′(x)y1(x)2 �� felhasználásával és pl. C = 1 választásával most 1 1 _c[′](x) =_ _√_ cos[2][ √]x _x,_ tehát _c(x) = 2 tan_ _[√]x._ Így y2(x) = 2 sin _[√]x, és az általános megoldás A cos_ _[√]x + B sin_ _[√]x._ 3. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) y[(4)] _−_ 2y[′′′] _−_ _y[′′]_ + 2y[′] = 0 b) y[′′] _−_ 4y[′] + 4y = 0 c) y[′′′] + 3y[′′] + 3y[′] + y = 0 d) y[′′] _−_ 4y[′] + 29y = 0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lineáris diﬀerenciálegyenlet megoldásterének bázisát alkotját. Elsőrendű egyenletrendszerré</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakítva</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> bevezetésével az egyenlet</span></p> <p style="top:134.2pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:91.4pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:108.9pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:98.5pt;left:169.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:98.5pt;left:246.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:98.5pt;left:323.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:98.5pt;left:369.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:98.5pt;left:421.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:117.9pt;left:169.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:117.9pt;left:246.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:117.9pt;left:323.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:111.0pt;left:370.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:110.0pt;left:422.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:129.4pt;left:170.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:129.4pt;left:247.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:130.4pt;left:320.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:130.4pt;left:370.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">...</span></p> <p style="top:137.4pt;left:421.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:151.8pt;left:169.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:151.8pt;left:246.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:151.8pt;left:323.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:151.8pt;left:369.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:151.8pt;left:421.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:168.1pt;left:138.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:158.1pt;left:148.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:175.4pt;left:154.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:158.1pt;left:158.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:168.1pt;left:164.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:168.1pt;left:215.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:158.1pt;left:225.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:175.4pt;left:231.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:158.1pt;left:235.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:168.1pt;left:241.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:168.1pt;left:292.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:158.1pt;left:302.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:175.4pt;left:308.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:158.1pt;left:312.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:168.1pt;left:318.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:168.1pt;left:369.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">· · ·</span></i></p> <p style="top:168.1pt;left:393.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:158.1pt;left:402.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> k</span></i></p> <p style="top:175.4pt;left:408.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:158.1pt;left:424.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:168.1pt;left:430.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:91.4pt;left:455.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:108.9pt;left:455.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:134.2pt;left:464.0pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:194.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lesz, tehát a Wronski-determinánsra a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (Tr</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> kλW</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet teljesül,</span></p> <p style="top:209.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">emiatt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kλx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:228.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. A Wronski-determináns segítségével határozzuk meg a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span></p> <p style="top:242.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldását, ha tudjuk, hogy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldja az egyenletet.</span></p> <p style="top:261.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet ekvivalens az</span></p> <p style="top:295.6pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:279.7pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:286.7pt;left:150.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:286.7pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:302.6pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:309.9pt;left:148.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:302.6pt;left:179.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:309.9pt;left:190.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></p> <p style="top:279.7pt;left:211.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:295.6pt;left:219.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:328.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elsőrendű egyenletrendszerrel, ennek Wronski-determinánsa teljesíti a</span></p> <p style="top:360.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:368.2pt;left:148.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i></sup></p> <p style="top:393.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletet. Most</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:400.5pt;left:353.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup></p> <p style="top:391.7pt;left:469.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">C</span></i></p> <p style="top:394.8pt;left:467.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:407.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Másrészt ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az eredeti egyenlet két megoldása, akkor a</span></p> <p style="top:439.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:423.4pt;left:150.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:432.5pt;left:153.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:432.5pt;left:190.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:446.9pt;left:153.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:453.1pt;left:159.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:446.9pt;left:190.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:453.1pt;left:195.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:423.4pt;left:216.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:473.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvény ugyanezt a diﬀerenciálegyenletet teljesíti. Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ismert, akkor ez</span></p> <p style="top:487.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re nézve elsőrendű lineáris inhomogén diﬀerenciálegyenletet jelent, amit az állandók</span></p> <p style="top:502.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">variálásának módszerével oldhatunk meg.</span></p> <p style="top:502.0pt;left:303.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A konstrukció miatt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldja a homogén</span></p> <p style="top:516.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletet, tehát az inhomogén egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban keressük.</span></p> <p style="top:532.9pt;left:88.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:541.9pt;left:91.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:541.9pt;left:134.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:556.4pt;left:91.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:562.5pt;left:97.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:556.4pt;left:128.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:532.9pt;left:186.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:532.9pt;left:206.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:541.9pt;left:209.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:541.9pt;left:277.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:556.4pt;left:209.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:562.5pt;left:215.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:556.4pt;left:245.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:562.5pt;left:272.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:532.9pt;left:356.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:532.9pt;left:375.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:541.9pt;left:379.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:541.9pt;left:437.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:556.4pt;left:379.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:562.5pt;left:384.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:556.4pt;left:415.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:532.9pt;left:465.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:581.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felhasználásával és pl.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> választásával most</span></p> <p style="top:611.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:603.0pt;left:163.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:619.3pt;left:147.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> √</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:603.0pt;left:193.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:610.7pt;left:188.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></p> <p style="top:642.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span></p> <p style="top:667.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2 tan</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></p> <p style="top:692.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Így</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2 sin</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, és az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:711.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását.</span></p> <p style="top:726.3pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></p> <p style="top:742.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:759.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:775.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 29</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_295.png	lincáris differenciálegyenlet megoldáste alakítva y — (yy7.. . 44-3) bevezi ének bázisát alkotját. Elsőr óvel az egyenlet egyenletrendszerré 0 1 0 [ 0 0 1 v [] 0 0 0 1 04 -P- Y (8JÜN, Josz, tehát a Wronski-detersninánsra a W" — (TeAJW — £XWV dífforenciálegyenlet teljesül emiatt W(2) — ePzW(0) 2. A Wronski-detet ével határozzuk meg a 4ry"--2414-y — 0 dífferenciálegyenlet általános megoldását, ha tudjuk, hogy cos /7 megoldja az egyenletet. Megoldás. Az azlzhy" 4 a(zy 4 aslzhy — 0 differe [ 0 1 ETÁNE váns segítsé elsőrendű egyenletrendszerrel, ennek Wronski-determinánsa teljesíti a 2, W) üetet. Most ay(2) — 2 az(e) — 4z, tehát W" — —L. amiből W() — £. Miásrészt ha 9 és 99 az eredoti egyenlet két megoldása; akkor a e) velz)] elo) vlz)) 4 a differenciálegyenletet teljesíti. Ha yi(r) — cos /7 ismert, akkor ez endűi lineáris inhomogén dítlerenciálegyenletet j. variálásának módszerével oldhatunk meg. A koönstrukció miatt ) megoldja a homogén aletet, tehát az inhomogén egyenlet megoldását elzjy,(z) alakban keressük. t, amit az állandók m e) Múle) elejviln) 4 elentől e) — elrlnle) ml lelejmidyi felhasználásával és pl. 1 1 (e0s? YT VT" h — 0 Th elenta) -nr 1 választásával most e tehát ele) — 2tan /Z Így vala) — 2sin /7. és az általános megoldás A cos /7 -- Bsin /7. 3. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) 92" 20 1) - 4y449—0 CE ÉZ AR ATT d) — 4y 4. 209— 0
e) y(4) + 2y′′ + y = 0 Megoldás. Az egyenletekbe az eλx függvényt helyettesítünk és elosztjuk mindkét oldalt ugyanezzel a függvénnyel. A bal oldal így λ egy polinomja lesz (karakterisztikus polinom), ha ennek λi gyöke mi multiplicitással, akkor az eλix, xeλix, . . . , xmi−1eλix függvények megoldják a diﬀerenciálegyenletet. Ez pontosan annyi függvényt határoz meg, mint amennyi az egyenlet rendje, és a függvények a megoldástér egy bázisát alkotják. (Ha λ = a+bi ∈C\R gyök m multiplicitással, akkor λ is az ugyanilyen multiplicitással, ezek alkalmas lineáris kombinációi valósak: eax cos bx, . . . , xm−1eax cos bx, eax sin bx, . . . , xm−1eax sin bx) , , , , , ) a) A karakterisztikus polinom λ4 −2λ3 −λ2 + 2λ = (λ + 1)λ(λ −1)(λ −2), az általános a) A karakterisztikus polinom λ 2λ λ + 2λ (λ + 1)λ(λ 1)(λ 2), az általános megoldás y(x) = Ae−x + B + Cex + De2x. b) A karakterisztikus polinom λ2 −4λ + 4 = (λ −2)2, az általános megoldás y(x) = b) A karakterisztikus polinom λ 4λ + 4 (λ 2) , az általános megoldás y(x) Ae2x + Bxe2x. c) A karakterisztikus polinom λ3 + 3λ2 + 3λ + 1 = (λ + 1)3, az általános megoldás y(x) = c) a a te s t us po o λ + 3λ + 3λ + (λ + ) Ae−x + Bxe−x + Cx2e−x. d) A karakterisztikus polinom λ2 −4λ + 29, ennek gyökei 4 ± √ 42 −4 · 29 = 2 ± √ −25 = 2 ± 5i, az általános megoldás y(x) = Ae2x cos 5x + Be2x sin 5x. e) A karakterisztikus polinom λ4 + 2λ2 + 1 = (λ2 + 1)2 = (λ + i)2(λ −i)2, az általános megoldás y(x) = A cos x + Bx cos x + C sin x + Dx sin x. 4. Legyenek ω ≥0 és α ≥0 valós paraméterek. Oldjuk meg az y′′ + 2αy′ + ω2y = 0 diﬀe- renciálegyenletet y(0) = 1, y′(0) = 0 kezdeti feltétel mellett. Miben különbözik a megoldás α > ω és α < ω esetén? Megoldás. A karakterisztikus polinom λ2 + 2αλ + ω2λ, ennek gyökei −2α ± √ 4α2 −4ω2 = −α ± √ α2 −ω2. Ha α > ω, akkor mindkét gyök valós (negatív), az általános megoldás √ y(x) = Ae(−α+ √ α2−ω2)x + Be(−α− √ α2−ω2)x. A kezdeti feltételből kell A és B értékét meghatározni: 1 = y(0) = A + B 0 = y′(0) = (−α + √ α2 −ω2)A + (−α − √ α2 −ω2)B, ebből √ A = α + 2 √ α2 −ω2 √ α2 −ω2 √ B = −α + 2 √ α2 −ω2 √ α2 −ω2 √ A kapott megoldás monoton csökken, exponenciálisan 0-hoz tart (y(x) ≤Ce(−α+ α2−ω2)x). Ha α < ω, akkor viszont komplex gyököket kapunk, az általános megoldás y(x) = Ae−αx cos( √ ω2 −α2x) + Be−αx sin( √ ω2 −α2x).	e) y[(4)] + 2y[′′] + y = 0 _Megoldás. Az egyenletekbe az e[λx]_ függvényt helyettesítünk és elosztjuk mindkét oldalt ugyanezzel a függvénnyel. A bal oldal így λ egy polinomja lesz (karakterisztikus polinom), ha ennek λi gyöke mi multiplicitással, akkor az e[λ][i][x], xe[λ][i][x], . . ., x[m][i][−][1]e[λ][i][x] függvények megoldják a differenciálegyenletet. Ez pontosan annyi függvényt határoz meg, mint amennyi az egyenlet rendje, és a függvények a megoldástér egy bázisát alkotják. (Ha λ = a + _bi ∈_ _\_ C R gyök m multiplicitással, akkor λ is az ugyanilyen multiplicitással, ezek alkalmas lineáris kombinációi valósak: e[ax] cos bx, . . ., x[m][−][1]e[ax] cos bx, e[ax] sin bx, . . ., x[m][−][1]e[ax] sin bx) a) A karakterisztikus polinom λ[4] _−_ 2λ[3] _−_ _λ[2]_ + 2λ = (λ + 1)λ(λ − 1)(λ − 2), az általános megoldás y(x) = Ae[−][x] + B + Ce[x] + De[2][x]. b) A karakterisztikus polinom λ[2] _−_ 4λ + 4 = (λ − 2)[2], az általános megoldás y(x) = _Ae[2][x]_ + Bxe[2][x]. c) A karakterisztikus polinom λ[3] + 3λ[2] + 3λ + 1 = (λ + 1)[3], az általános megoldás y(x) = _Ae[−][x]_ + Bxe[−][x] + Cx[2]e[−][x]. d) A karakterisztikus polinom λ[2] _−_ 4λ + 29, ennek gyökei _√_ 4 ± 4[2] _−_ 4 · 29 _√_ = 2 ± 2 _−25 = 2 ± 5i,_ az általános megoldás y(x) = Ae[2][x] cos 5x + Be[2][x] sin 5x. e) A karakterisztikus polinom λ[4] + 2λ[2] + 1 = (λ[2] + 1)[2] = (λ + i)[2](λ − _i)[2], az általános_ megoldás y(x) = A cos x + Bx cos x + C sin x + Dx sin x. 4. Legyenek ω ≥ 0 és α ≥ 0 valós paraméterek. Oldjuk meg az y[′′] + 2αy[′] + ω[2]y = 0 differenciálegyenletet y(0) = 1, y[′](0) = 0 kezdeti feltétel mellett. Miben különbözik a megoldás _α > ω és α < ω esetén?_ _Megoldás. A karakterisztikus polinom λ[2]_ + 2αλ + ω[2]λ, ennek gyökei _√_ _−2α ±_ 4α[2] _−_ 4ω[2] _√_ = −α ± 2 _α[2]_ _−_ _ω[2]._ Ha α > ω, akkor mindkét gyök valós (negatív), az általános megoldás _√_ _√_ _y(x) = Ae[(][−][α][+]_ _α[2]−ω[2])x + Be(−α−_ _α[2]−ω[2])x._ A kezdeti feltételből kell A és B értékét meghatározni: 1 = y(0) = A + B _√_ 0 = y[′](0) = (−α + _√_ _α[2]_ _−_ _ω[2])A + (−α −_ _α[2]_ _−_ _ω[2])B,_ ebből _√_ _α[2]_ _−_ _ω[2]_ _A =_ _[α][ +]√_ 2 _α[2]_ _−_ _ω[2]_ _√_ _α[2]_ _−_ _ω[2]_ _B =_ _[−][α][ +]√_ _._ 2 _α[2]_ _−_ _ω[2]_ _√_ A kapott megoldás monoton csökken, exponenciálisan 0-hoz tart (y(x) ≤ _Ce[(][−][α][+]_ _α[2]−ω[2])x)._ Ha α < ω, akkor viszont komplex gyököket kapunk, az általános megoldás _ω[2]_ _−_ _α[2]x)._ _√_ _y(x) = Ae[−][αx]_ cos( _√_ _ω[2]_ _−_ _α[2]x) + Be[−][αx]_ sin( -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:78.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenletekbe az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvényt helyettesítünk és elosztjuk mindkét oldalt</span></p> <p style="top:93.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ugyanezzel a függvénnyel. A bal oldal így</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egy polinomja lesz (karakterisztikus polinom),</span></p> <p style="top:107.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha ennek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gyöke</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> multiplicitással, akkor az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λ</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">i</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λ</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">i</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">i</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λ</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">i</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvények meg-</span></p> <p style="top:121.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">oldják a diﬀerenciálegyenletet. Ez pontosan annyi függvényt határoz meg, mint amennyi az</span></p> <p style="top:136.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet rendje, és a függvények a megoldástér egy bázisát alkotják. (Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">bi</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">C</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span></p> <p style="top:150.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">gyök</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> multiplicitással, akkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> is az ugyanilyen multiplicitással, ezek alkalmas lineáris</span></p> <p style="top:165.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kombinációi valósak:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ax</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> bx, . . . , x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ax</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> bx, e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ax</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> bx, . . . , x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ax</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> bx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:179.6pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános</span></p> <p style="top:194.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> De</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:210.5pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:225.0pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ae</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:241.4pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:255.8pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:272.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 29</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ennek gyökei</span></p> <p style="top:299.0pt;left:128.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:289.2pt;left:148.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:299.0pt;left:158.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 29</span></p> <p style="top:315.3pt;left:166.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:307.1pt;left:214.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:297.2pt;left:247.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:307.1pt;left:257.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">25 = 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i,</span></i></p> <p style="top:337.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:354.1pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános</span></p> <p style="top:368.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Dx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:388.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀe-</span></p> <p style="top:402.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">renciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Miben különbözik a megoldás</span></p> <p style="top:416.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α > ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α < ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén?</span></p> <p style="top:436.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αλ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ennek gyökei</span></p> <p style="top:462.3pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:452.5pt;left:144.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:462.3pt;left:154.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:478.6pt;left:153.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:470.4pt;left:210.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:460.0pt;left:254.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:470.4pt;left:264.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:500.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α > ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor mindkét gyök valós (negatív), az általános megoldás</span></p> <p style="top:528.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span></sup></p> <p style="top:520.4pt;left:180.2pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:527.1pt;left:187.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></p> <p style="top:520.4pt;left:272.9pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:527.1pt;left:280.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:555.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltételből kell</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékét meghatározni:</span></p> <p style="top:581.6pt;left:109.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></p> <p style="top:601.3pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:590.9pt;left:203.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:601.3pt;left:213.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:590.9pt;left:316.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:601.3pt;left:326.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B,</span></i></p> <p style="top:627.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ebből</span></p> <p style="top:659.5pt;left:107.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:641.6pt;left:154.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:651.4pt;left:164.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:668.6pt;left:140.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:658.8pt;left:146.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:668.6pt;left:156.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:693.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:675.2pt;left:164.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:685.0pt;left:174.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:702.3pt;left:145.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:692.5pt;left:151.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:702.3pt;left:161.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:693.1pt;left:214.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:728.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kapott megoldás monoton csökken, exponenciálisan</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-hoz tart (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span></sup></p> <p style="top:720.8pt;left:489.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:727.5pt;left:496.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">).</span></p> <p style="top:743.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α < ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor viszont komplex gyököket kapunk, az általános megoldás</span></p> <p style="top:769.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span></p> <p style="top:759.1pt;left:197.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:769.5pt;left:207.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin(</span></p> <p style="top:759.1pt;left:325.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:769.5pt;left:335.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_296.png	DESETVZT] Megoldás. Az egyenletekbe az €" függvényt helyettesítünk és elosztjuk mindkét oldalt gyanczzel a fűggvénnyel. A bal oldal így A egy polinomja lesz (karakterisztikus polinom), ha ennek A, győke m, multiplicitással, akkor az M, zel, ,,, , 2. a) A karakterisztikus polinom X8— 218- 32 4.29 — (X 4 1403 — 1)(3 — 2), az általános megoldás yíz) — Ae-? 4. B.-4-Ce" 4 De. 1) A karakterisztikus polinom 9? — 4X 4. 4 AZ" 4 Br" e) A karakterisztikus polinom 3? 433 4-33 41 — (X4-199, az általános m Aet a Bzet 4 Crtet. d) A karakterisztikus polinom 9? — 4X 4 29. ennek győkei ETVZETTE :] (3 — 2Y.. az általános megoldás yír) toldás yíz) — s2aY-B5-225i az általános megoldás y(z) — Ae: cos5z 4 Bettsin5r. e) A karakterisztikus polinom X! 4222 41 — ( 4 192 — (X -4-GY2(X — 09, az általános megoldás yíz) — Acosz 4 Bzcosz 4 Csinz 4 Drsint. Legyenek a 2 0 és a 2 0 valós paraméterek. Oldjuk meg az y" 4. 2a9 4- ! renclálegyentetet 9í0) — 1. 4/(0) — 0 kezdeti feltétel mellett. Miben külö díffe .Megoldás, A karakterisztikus polinom 9? 4- 2a -Fss2), ennek gyökel 2a 4 VIGTZ 2 a Var Ha a — 2, akkor mindkét győk valós (negatív az általános megoldás wíz) a AAEE 4 Bel-a-VA , A kezdeti feltételből kell A és 5 ér meghatározni: z0-4458 0-Y(0) — (-a 4 VAZA 4 (-a — VAZ BB, ebből FELENCET:] Er A kapott megoldás monoton csökken, exponenciálisan 0-hoz tart (y(r) £ Cel-t. Ha a 2 e), akkor viszont komplex győköket kapunk, az általános megoldás wía) — AT" cos( VOT alz) 4 Be-tt sin( VOE í. ]
A kezdeti feltételből kell A és B értékét meghatározni: 1 = y(0) = A 0 = y′(0) = −αA + √ ω2 −α2B, ebből A = 1 B = α √ ω2 −α2. Ilyenkor a megoldás a 0 körül oszcillál. Ha α > 0, akkor exponenciálisan 0-hoz tart (\|y(x)\| ≤ Ce−αx), ha viszont α = 0, akkor periodikus. Meg kell még vizsgálni az α = ω esetet. Ekkor −α kétszeres valós gyök, az általános megoldás y(x) = Ae−αx + Bxe−αx. A kezdeti feltételből kell A és B értékét meghatározni: 1 = y(0) = A 0 = y′(0) = −αA + B, ebből A = 1 és B = α. Ekkor a megoldás szigorúan monoton csökken, szintén exponenciálisan 0-hoz tart, de kicsivel lassabban mint e−αx (\|y(x)\| ≤Cxe−αx). Ezt a megoldást az előző két eset bármelyikéből megkaphattuk volna ω →α határátmenettel. 5. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását. g a) y′′ −4y′ −12y = xex ) y y y b) y′′ −4y′ −12y = xe−2x ) c) y′′′ −4y′′ + 4y′ = x2 + e2x ) y y y d) y′′ −2y′ + 5y = ex sin 2x Megoldás. Ha az inhomogén tag p(x)eλx alakú, ahol p(x) egy d fokú polinom és λ a karakterisztikus polinomnak m-szeres gyöke (m = 0 ha nem gyök), akkor az egyenletnek létezik q(x)xmeλx alakú megoldása, ahol q(x) szintén d fokú polinom. (Komplex gyök esetén cos, sin előállítható a komplex exponenciálisokból.) s e őá t ató a o p e e po e c á so bó ) a) A karakterisztikus polinom λ2 −4λ−12 = (λ+2)(λ−6), a homogén egyenlet általános p ( + )( ), g gy megoldása y(x) = Ae−2x + Be6x. Nincsen rezonancia, (C0 + C1x)ex alakú megoldást keresünk. Ezt behelyettesítve az xex = 2C1ex + (C0 + C1x)ex −4(C1ex + (C0 + C1x)ex) −12(C0 + C1x)ex = (−15C0 −2C1)ex + (−15C1)xex xex = 2C1ex + (C0 + C1x)ex −4(C1ex + (C0 + C1x)ex) −12(C0 + C1x)ex egyenlethez jutunk, ami akkor teljesül minden x-re, ha C1 = −1 15, C2 = 2 225. Tehát az általános megoldás egyenlethez jutunk, ami akkor teljesül minden x-re, ha C1 = −1 15 y(x) = 2 225ex −1 15xex + Ae−2x + Be6x. y(x) = 2 225ex −1 15 b) A karakterisztikus polinom λ2 −4λ−12 = (λ+2)(λ−6), a homogén egyenlet általános megoldása y(x) = Ae−2x + Be6x. Külső rezonancia van, (C0 + C1x)xe−2x = (C0x + C1x2)e−2x alakú megoldást keresünk. Ezt behelyettesítve az b) A karakterisztikus polinom λ2 −4λ−12 = (λ+2)(λ−6), a homogén egyenlet általános (−8C0 + 2C1)e−2x −16C1xe−2x = xe−2x egyenlet adódik, ebből C1 = −1 16 64 1 16 és C2 = −1 1 64, így az általános megoldás y(x) = −1 64 1 64xe−2x −1 16 1 16x2e−2x + Ae−2x + Be6x.	A kezdeti feltételből kell A és B értékét meghatározni: 1 = y(0) = A _√_ 0 = y[′](0) = −αA + _ω[2]_ _−_ _α[2]B,_ ebből _A = 1_ _α_ _B =_ _√_ _ω[2]_ _−_ _α[2]_ _[.]_ Ilyenkor a megoldás a 0 körül oszcillál. Ha α > 0, akkor exponenciálisan 0-hoz tart (\|y(x)\| ≤ _Ce[−][αx]), ha viszont α = 0, akkor periodikus._ Meg kell még vizsgálni az α = ω esetet. Ekkor −α kétszeres valós gyök, az általános megoldás y(x) = Ae[−][αx] + Bxe[−][αx]. A kezdeti feltételből kell A és B értékét meghatározni: 1 = y(0) = A 0 = y[′](0) = −αA + B, ebből A = 1 és B = α. Ekkor a megoldás szigorúan monoton csökken, szintén exponenciálisan 0-hoz tart, de kicsivel lassabban mint e[−][αx] (\|y(x)\| ≤ _Cxe[−][αx]). Ezt a megoldást az_ előző két eset bármelyikéből megkaphattuk volna ω → _α határátmenettel._ 5. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) y[′′] _−_ 4y[′] _−_ 12y = xe[x] b) y[′′] _−_ 4y[′] _−_ 12y = xe[−][2][x] c) y[′′′] _−_ 4y[′′] + 4y[′] = x[2] + e[2][x] d) y[′′] _−_ 2y[′] + 5y = e[x] sin 2x _Megoldás. Ha az inhomogén tag p(x)e[λx]_ alakú, ahol p(x) egy d fokú polinom és λ a karakterisztikus polinomnak m-szeres gyöke (m = 0 ha nem gyök), akkor az egyenletnek létezik _q(x)x[m]e[λx]_ alakú megoldása, ahol q(x) szintén d fokú polinom. (Komplex gyök esetén cos, sin előállítható a komplex exponenciálisokból.) a) A karakterisztikus polinom λ[2] _−_ 4λ _−_ 12 = (λ +2)(λ _−_ 6), a homogén egyenlet általános megoldása y(x) = Ae[−][2][x] + Be[6][x]. Nincsen rezonancia, (C0 + C1x)e[x] alakú megoldást keresünk. Ezt behelyettesítve az _xe[x]_ = 2C1e[x] + (C0 + C1x)e[x] _−_ 4(C1e[x] + (C0 + C1x)e[x]) − 12(C0 + C1x)e[x] = (−15C0 − 2C1)e[x] + (−15C1)xe[x] egyenlethez jutunk, ami akkor teljesül minden x-re, ha C1 = − 15[1] [,][ C][2][ =] 2252 [. Tehát az] általános megoldás 2 _y(x) =_ 225[e][x][ −] 15[1] _[xe][x][ +][ Ae][−][2][x][ +][ Be][6][x][.]_ b) A karakterisztikus polinom λ[2] _−_ 4λ _−_ 12 = (λ +2)(λ _−_ 6), a homogén egyenlet általános megoldása y(x) = Ae[−][2][x] + Be[6][x]. Külső rezonancia van, (C0 + C1x)xe[−][2][x] = (C0x + _C1x[2])e[−][2][x]_ alakú megoldást keresünk. Ezt behelyettesítve az (−8C0 + 2C1)e[−][2][x] _−_ 16C1xe[−][2][x] = xe[−][2][x] egyenlet adódik, ebből C1 = − 16[1] [és][ C][2][ =][ −] 64[1] [, így az általános megoldás] _y(x) = −_ [1] 64[xe][−][2][x][ −] 16[1] _[x][2][e][−][2][x][ +][ Ae][−][2][x][ +][ Be][6][x][.]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltételből kell</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékét meghatározni:</span></p> <p style="top:82.8pt;left:109.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:102.5pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αA</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:92.1pt;left:207.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:102.5pt;left:217.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B,</span></i></p> <p style="top:126.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ebből</span></p> <p style="top:150.0pt;left:107.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></p> <p style="top:171.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:163.4pt;left:153.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:170.9pt;left:132.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:180.7pt;left:142.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:201.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ilyenkor a megoldás a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> körül oszcillál. Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor exponenciálisan</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-hoz tart (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\| ≤</span></i></p> <p style="top:216.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), ha viszont</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor periodikus.</span></p> <p style="top:230.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Meg kell még vizsgálni az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetet.</span></p> <p style="top:230.5pt;left:304.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ekkor</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kétszeres valós gyök, az általános</span></p> <p style="top:244.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A kezdeti feltételből kell</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értékét meghatározni:</span></p> <p style="top:268.7pt;left:109.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:286.1pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αA</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B,</span></i></p> <p style="top:309.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ekkor a megoldás szigorúan monoton csökken, szintén exponenci-</span></p> <p style="top:324.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">álisan</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-hoz tart, de kicsivel lassabban mint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\| ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Cxe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">αx</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">). Ezt a megoldást az</span></p> <p style="top:338.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">előző két eset bármelyikéből megkaphattuk volna</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> határátmenettel.</span></p> <p style="top:357.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását.</span></p> <p style="top:371.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:387.9pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:404.4pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:420.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:439.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ha az inhomogén tag</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> fokú polinom és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a karak-</span></p> <p style="top:453.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">terisztikus polinomnak</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-szeres gyöke (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ha nem gyök), akkor az egyenletnek létezik</span></p> <p style="top:468.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">q</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">λx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú megoldása, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> q</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szintén</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> d</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> fokú polinom. (Komplex gyök esetén</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:482.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> előállítható a komplex exponenciálisokból.)</span></p> <p style="top:496.9pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet általános</span></p> <p style="top:511.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Nincsen rezonancia,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú megoldást</span></p> <p style="top:525.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">keresünk. Ezt behelyettesítve az</span></p> <p style="top:548.5pt;left:128.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:565.9pt;left:149.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:590.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlethez jutunk, ami akkor teljesül minden</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:597.7pt;left:423.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">15</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:588.9pt;left:475.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:597.7pt;left:470.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">225</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Tehát az</span></sup></p> <p style="top:604.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldás</span></p> <p style="top:633.1pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:625.0pt;left:171.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:641.3pt;left:165.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">225</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:641.3pt;left:211.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:661.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet általános</span></p> <p style="top:675.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Külső rezonancia van,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></p> <p style="top:689.9pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú megoldást keresünk. Ezt behelyettesítve az</span></p> <p style="top:713.7pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:737.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet adódik, ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:744.7pt;left:255.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">16</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:744.7pt;left:322.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">64</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, így az általános megoldás</span></sup></p> <p style="top:767.4pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:775.6pt;left:175.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">64</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:775.6pt;left:231.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">6</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_297.png	A kezdeti felt w(0) /(0) — —aA 4 VT 2B, telből kell A és B éri meghatározni: ebből Ilyenkor a megoldás a 0 körül oszcillál. Ha a - 0, akkor exponenciálisan 0-hoz tart ([y(7)] £ (Ce-"7), ha viszont a — 0, akkor periodikus. Meg kell móg vizsgálni az a — 2s esetet. Ekkor —a kétszeres valós gyök, az általános megoldás y(z) — Ae"" 4 Bze""". A közdeti feltételből kell A és B értékét meghatározni: w(0) v(0)- aA 4 58. [ 5. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) 4 l9r 1) 1— 4y — 129 — rels AY ády ét d) — 2y 4. őy — Esin2r a) A karakterisztikus polinom 32 —41 — 12 keresünk. Ezt behelyettesítve az nE 4 (Ca 4 CrE — MCE 4 (Ca F OZJE ) — 12(Co 4 CizJE 1504—20)E 4 (-150Y2E et általános rezonancia, (Cy 4. Ciz)e" alakú megoldást egyenlethez jutunk, ami akkor teljesül minden 2-re, ha C1 Tehát az általános megoldás wíz) — 755 1) A karakterisztikus polinom 32 —43-—12 Cya2)e-2: alakú megoldást keresünk. Ezt behelyettesítve az et általános Cor 4 (-8C9 4 2YE — 16047E — egyenlet adódik, ebből C, — —k és C — F 4 AT 4 BE
c) A karakterisztikus polinom λ3 −4λ2 + 4λ = λ(λ −2)2, a homogén egyenlet általános megoldása y(x) = A + Be2x + Cxe2x (belső rezonancia). Mindkét inhomogén tag miatt külső rezonancia van, az inhomogén egyenlet megoldását C0x + C1x2 + C2x3 + D0x2e2x alakban keressük. Behelyettesítve: (4C0 −8C1 + 6C2) + (8C1 −24C2)x + 12C2x2 + 4D0e2x = x2 + e2x, amiből D0 = 1 4 1 4, C2 = 1 12, C1 = 1 4 1 4, C0 = 3 8 3 8. Az általános megoldás y(x) = 3 8 3 8x + 1 4 1 4x2 + 1 12 1 12x3 + 1 4 1 4x2e2x + A + Be2x + Cxe2x. d) A karakterisztikus polinom λ2−2λ+5 = (λ−1+2i)(λ−1−2i), a homogén egyenlet ál- talános megoldása y(x) = Aex cos 2x+Bex sin 2x. Külső rezonancia van, az inhomogén egyenlet megoldását C0xex cos 2x + D0xex sin 2x alakban kereshetjük. Behelyettesítve: d) A karakterisztikus polinom λ2−2λ+5 = (λ−1+2i)(λ−1−2i), a homogén egyenlet ál- 4D0ex cos 2x −4C0ex sin 2x = ex sin 2x, tehát C0 = −1 4 és D0 = 0. Az általános megoldás y(x) = −1 4xex cos 2x + Aex cos 2x + Bex sin 2x. További gyakorló feladatok 6. Bizonyítsuk be, hogy az y′ 1 y′ 1 = y2 y′ 2 = −e y′ 2 = −e2xy1 + y2 diﬀerenciálegyenlet-rendszernek létezik nem korlátos megoldása. Megoldás. Az egyenletrendszer mátrix alakban y′ = A(x)y, ahol A(x) = . −e2x 1 Két lineárisan független megoldásból mátrixot képezhetünk, ennek determinánsa a W(x) Wronski-determináns. A W ′(x) = Tr A(x)W(x) = W(x) egyenlet megoldása W(x) = Cex, ahol C ̸= 0 állandó. Ha minden megoldás korlátos lenne, akkor a belőlük képzett determináns is korlátos lenne, de ex nem az. Tehát létezik nem korlátos megoldás. 7. Határozzuk meg az y′′ −y′ −e2xy = 0 diﬀerenciálegyenlet általános megoldását, ha tudjuk, hogy eex megoldás. Megoldás. A Wronski-determinánsra W ′ = W teljesül, tehát W = Cex. y1(x) = eex Két lineárisan független megoldásból mátrixot képezhetünk, ennek determinánsa a W(x) Wronski-determináns. A W ′(x) = Tr A(x)W(x) = W(x) egyenlet megoldása W(x) = Cex, ahol C ̸= 0 állandó. Ha minden megoldás korlátos lenne, akkor a belőlük képzett determináns is korlátos lenne, de ex nem az. Tehát létezik nem korlátos megoldás. 7. Határozzuk meg az y′′ −y′ −e2xy = 0 diﬀerenciálegyenlet általános megoldását, ha tudjuk, megoldás, egy másikat y2(x) = c(x)y1(x) alakban keressük (C = 1 feltehető): y1(x) c(x)y1(x) y′ 1(x) (c(x)y1(x))′ y1(x) c(x)y1(x) y′ 1(x) (c(x)y1(x)) = c′(x)y1(x)2, ex = tehát c′(x) = ex e2ex . Ezt integrálva c(x) = −1 2e−2ex + C adódik, az általános megoldás y(x) = C1eex + C2e−ex.	c) A karakterisztikus polinom λ[3] _−_ 4λ[2] + 4λ = λ(λ − 2)[2], a homogén egyenlet általános megoldása y(x) = A + Be[2][x] + Cxe[2][x] (belső rezonancia). Mindkét inhomogén tag miatt külső rezonancia van, az inhomogén egyenlet megoldását C0x + C1x[2] + C2x[3] + D0x[2]e[2][x] alakban keressük. Behelyettesítve: (4C0 − 8C1 + 6C2) + (8C1 − 24C2)x + 12C2x[2] + 4D0e[2][x] = x[2] + e[2][x], amiből D0 = [1]4 [,][ C][2][ =] 121 [,][ C][1][ =][ 1]4 [,][ C][0][ =][ 3]8 [. Az általános megoldás] _y(x) = [3]_ 8[x][ + 1]4[x][2][ + 1]12[x][3][ + 1]4[x][2][e][2][x][ +][ A][ +][ Be][2][x][ +][ Cxe][2][x][.] d) A karakterisztikus polinom λ[2] _−2λ+5 = (λ−1+2i)(λ−1−2i), a homogén egyenlet ál-_ talános megoldása y(x) = Ae[x] cos 2x + _Be[x]_ sin 2x. Külső rezonancia van, az inhomogén egyenlet megoldását C0xe[x] cos 2x + D0xe[x] sin 2x alakban kereshetjük. Behelyettesítve: 4D0e[x] cos 2x − 4C0e[x] sin 2x = e[x] sin 2x, tehát C0 = − [1]4 [és][ D][0][ = 0. Az általános megoldás] _y(x) = −[1]_ 4[xe][x][ cos 2][x][ +][ Ae][x][ cos 2][x][ +][ Be][x][ sin 2][x.] ## További gyakorló feladatok 6. Bizonyítsuk be, hogy az _y1[′]_ [=][ y][2] _y2[′]_ [=][ −][e][2][x][y][1] [+][ y][2] differenciálegyenlet-rendszernek létezik nem korlátos megoldása. _Megoldás. Az egyenletrendszer mátrix alakban y[′]_ = A(x)y, ahol _A(x) =_ � 0 1� _._ _−e[2][x]_ 1 Két lineárisan független megoldásból mátrixot képezhetünk, ennek determinánsa a W (x) Wronski-determináns. A W _[′](x) = Tr A(x)W_ (x) = W (x) egyenlet megoldása W (x) = _Ce[x], ahol C ̸= 0 állandó. Ha minden megoldás korlátos lenne, akkor a belőlük képzett_ determináns is korlátos lenne, de e[x] nem az. Tehát létezik nem korlátos megoldás. 7. Határozzuk meg az y[′′] _−_ _y[′]_ _−_ _e[2][x]y = 0 differenciálegyenlet általános megoldását, ha tudjuk,_ hogy e[e][x] megoldás. _Megoldás. A Wronski-determinánsra W_ _[′]_ = W teljesül, tehát W = Ce[x]. _y1(x) = e[e][x]_ megoldás, egy másikat y2(x) = c(x)y1(x) alakban keressük (C = 1 feltehető): = c′(x)y1(x)2, �� _e[x]_ = _y1(x)_ _c(x)y1(x)_ _y1[′]_ [(][x][)] (c(x)y1(x))[′] �� tehát _c[′](x) =_ _[e][x]_ _e[2][e][x][ .]_ Ezt integrálva _c(x) = −[1]_ 2[e][−][2][e][x][ +][ C] adódik, az általános megoldás _y(x) = C1e[e][x]_ + C2e[−][e][x]. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet általános</span></p> <p style="top:73.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cxe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(belső rezonancia). Mindkét inhomogén tag miatt</span></p> <p style="top:88.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">külső rezonancia van, az inhomogén egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:102.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük. Behelyettesítve:</span></p> <p style="top:123.8pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + (8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">24</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:145.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:152.4pt;left:167.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:143.6pt;left:212.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:152.4pt;left:210.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:152.4pt;left:257.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 3</span></sup></p> <p style="top:152.4pt;left:299.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">8</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános megoldás</span></sup></p> <p style="top:172.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup></p> <p style="top:180.9pt;left:165.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:180.9pt;left:195.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:180.9pt;left:229.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:180.9pt;left:269.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cxe</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:198.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+5 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet ál-</span></p> <p style="top:212.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">talános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Be</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Külső rezonancia van, az inhomogén</span></p> <p style="top:227.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban kereshetjük. Behelyettesítve:</span></p> <p style="top:248.5pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></p> <p style="top:269.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:277.1pt;left:167.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános megoldás</span></sup></p> <p style="top:297.4pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:305.6pt;left:175.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></sup></p> <p style="top:330.5pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:354.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Bizonyítsuk be, hogy az</span></p> <p style="top:375.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:382.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:393.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:399.5pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:414.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszernek létezik nem korlátos megoldása.</span></p> <p style="top:431.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenletrendszer mátrix alakban</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span></p> <p style="top:460.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:444.8pt;left:146.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:453.4pt;left:161.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:453.4pt;left:186.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:467.8pt;left:152.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:467.8pt;left:186.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:444.8pt;left:192.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:460.7pt;left:200.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:489.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Két lineárisan független megoldásból mátrixot képezhetünk, ennek determinánsa a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:504.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Wronski-determináns.</span></p> <p style="top:504.0pt;left:199.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = Tr</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:518.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> állandó. Ha minden megoldás korlátos lenne, akkor a belőlük képzett</span></p> <p style="top:532.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">determináns is korlátos lenne, de</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nem az. Tehát létezik nem korlátos megoldás.</span></p> <p style="top:550.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet általános megoldását, ha tudjuk,</span></p> <p style="top:564.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás.</span></p> <p style="top:581.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> A Wronski-determinánsra</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> teljesül, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:581.8pt;left:475.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup></p> <p style="top:596.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás, egy másikat</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban keressük (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> feltehető):</span></p> <p style="top:625.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:608.7pt;left:132.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:617.7pt;left:136.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:617.7pt;left:178.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:632.2pt;left:136.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:638.3pt;left:141.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:632.2pt;left:172.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:608.7pt;left:231.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:653.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span></p> <p style="top:676.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:685.1pt;left:147.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> .</span></i></sup></p> <p style="top:701.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ezt integrálva</span></p> <p style="top:727.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:735.2pt;left:153.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup></p> <p style="top:751.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik, az általános megoldás</span></p> <p style="top:772.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_298.png	€) A karakterisztikus polinom 3? — 432 4-4) — M(X — 272, a homogén egyenlet általános 40 m koc s 24(',); 41202 4 4DÉ amiből Do — teE 4 A 4 BE 4 Czé. d) A karakterisztikus polinom 32—214-5 — (1—14-2/(3— 1-2i), a homogén egyenlet ál talános megoldása yír) — Ae" cos2r--Besin 2r. Külső rezonancia van, az inhomogé egyenlet megoldását Chre" cos2r 4. Duze" sin 2x alakban kereshetjűk. Behelyettesítve AD 0082x — 4Cye sin2r — €" sin2z, tehát C — —1 és Dy —0. Az általános megoldás 2 gejcos2z 4 A" cos2z 4 BE W) — gz€ cos2z 4 Ar cos2z 4 BE sin 2r "További gyakorló feladatok ú FA differenciálegyer m RT dszernek létezik nem korlátos megoldása. A(zjy, ahol "Megoldás. Az egyen a0-[2. ] Két lincárisan fűggetlen megokdásból mátrixot képezhetüűnk, e MWronski-determináns. A W(z) — Tr A(zjW(r) — IW(z) egyenlet megoldása W(z) — Ce, ahol C 2 0 állandó. Ha minden megoldás korlátos lenne, akkor a belőlük képzett determináns is korlátos lenne, de e" nem az. Tehát létezik nem korlátos megoldás. trendszer mátrix alakban Y" nek determini 1sa a W(z) - CE. ml e (z)m(z) alakban keressük (C — 1 feltehető) .Megoldás. A Wronski-determi megoldás, egy másikat ya(z) eh eon -[ telanteyi 7 7O e Ezt integrálva elz) adódik, az általános megoldás a) OE 4 Cse
8. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását. g a) y′′ + 2y′ + 10y = 0 b) y′′ −12y′ + 27y = 0 ) c) y′′ −10y′ + 25y = 0 d) y(4) + 18y′′ + 81y = 0 ) e) y′′′ −6y′′ + 12y′ −8y = 0 ) y y y y f) y(n) −y = 0, ahol n ≥1 egész Megoldás. a) A karakterisztikus polinom λ2+2λ+10 = (λ+1−3i)(λ+1+3i), az általános megoldás a) A karakterisztikus polinom λ +2λ+10 (λ+1 3i)(λ+1+3i), az általános megoldás y(x) = Ae−x cos 3x + Be−x sin 3x. b) A karakterisztikus polinom λ2 −12λ + 27 = (λ −3)(λ −9), az általános megoldás b) A karakterisztikus polinom λ 12λ + 27 (λ 3)(λ 9), az általános megoldás y(x) = Ae3x + Be9x. c) A karakterisztikus polinom λ2 −10λ + 25 = (λ −5)2, az általános megoldás y(x) = c) A karakterisztikus polinom λ 10λ + 25 (λ 5) , az általános megoldás y(x) Ae5x + Bxe5x. d) A karakterisztikus polinom λ4 +18λ2 +81 = (λ2 +9)2 = (λ+3i)2(λ−3i)2, az általános megoldás y(x) = A cos 3x + B sin 3x + Cx cos 3x + Dx sin 3x. e) A karakterisztikus polinom λ3 −6λ2 + 12λ −8 = (λ −2)3, az általános megoldás y(x) = Ae2x + Bxe2x + Cx2e2x. f) A karakterisztikus polinom megoldás y(x) = A cos 3x + B sin 3x + Cx cos 3x + Dx sin 3x. e) A karakterisztikus polinom λ3 −6λ2 + 12λ −8 = (λ −2)3, az általános megoldás n−1 λn −1 = n (λ −e2πi k k n). k=0 Ha n páratlan, akkor az általános megoldás n−1 y(x) = Aex + n Ckecos(2πi k n k n) cos esin(2πi k n k n) + Dkecos(2πi k n k n) sin esin(2πi k k n) k=1 ha n páros, akkor pedig n 2 −1 X y(x) = Aex + Be−x + n Ckecos(2πi k n k n) cos esin(2πi k n k n) + Dkecos(2πi k n k n) sin esin(2πi k k n) k=1 9. Legyenek ω0 > 0 és ω ≥0 valós paraméterek. Oldjuk meg az y′′ + ω2 0y = sin(ωx) diﬀerenciálegyenletet y(0) = 0, y′(0) = 0 kezdeti feltétel mellett. Mi történik, ha ω = ω0? Megoldás. Az egyenlet inhomogén lineáris, a hozzá tartozó homogén egyenlet y′′ + ω2 0 Megoldás. Az egyenlet inhomogén lineáris, a hozzá tartozó homogén egyenlet y′′ + ω2 0y. A karakterisztikus polinom λ2 + ω2 0 = (λ −iω0)(λ + iω0), tehát a homogén egyenlet általános karakterisztikus polinom λ2 + ω2 0 = (λ −iω0)(λ + iω0), tehát a homogén egyenlet általános megoldása y(x) = A cos(ω0x) + B sin(ω0x). Ha ω ̸= ω0, akkor nincsen külső rezonancia, az inhomogén egyenlet megoldását C cos(ωx) + D sin(ωx) alakban kereshetjük. Ezt behelyettesítve az egyenlet −ω2C cos(ωx) −ω2D sin(ωx) + ω2 0 ω2 0C cos(ωx) + ω2 0 ω2 0D sin(ωx) = sin(ωx), amiből C(ω2 0 ω2 0 0 −ω2) = 0, D(ω2 ω2 0 −ω2) = 1. Az általános megoldás y(x) = 1 ω2 0 −ω2 sin(ωx) + A cos(ω0x) + B sin(ω0x). y(x) = ω2 0 Az A, B paraméterek értékét úgy kell megválasztani, hogy a kezdeti feltétel teljesüljön. y′(x) = ω ω2 0 −ω2 cos(ωx) −Aω0 sin(ω0x) + Bω0 cos(ω0x)	8. Határozzuk meg az alábbi differenciálegyenletek általános megoldását. a) y[′′] + 2y[′] + 10y = 0 b) y[′′] _−_ 12y[′] + 27y = 0 c) y[′′] _−_ 10y[′] + 25y = 0 d) y[(4)] + 18y[′′] + 81y = 0 e) y[′′′] _−_ 6y[′′] + 12y[′] _−_ 8y = 0 f) y[(][n][)] _−_ _y = 0, ahol n ≥_ 1 egész _Megoldás._ a) A karakterisztikus polinom λ[2] +2λ+10 = (λ+1−3i)(λ+1+3i), az általános megoldás _y(x) = Ae[−][x]_ cos 3x + Be[−][x] sin 3x. b) A karakterisztikus polinom λ[2] _−_ 12λ + 27 = (λ − 3)(λ − 9), az általános megoldás _y(x) = Ae[3][x]_ + Be[9][x]. c) A karakterisztikus polinom λ[2] _−_ 10λ + 25 = (λ − 5)[2], az általános megoldás y(x) = _Ae[5][x]_ + Bxe[5][x]. d) A karakterisztikus polinom λ[4] +18λ[2] +81 = (λ[2] +9)[2] = (λ +3i)[2](λ _−_ 3i)[2], az általános megoldás y(x) = A cos 3x + B sin 3x + Cx cos 3x + Dx sin 3x. e) A karakterisztikus polinom λ[3] _−_ 6λ[2] + 12λ − 8 = (λ − 2)[3], az általános megoldás _y(x) = Ae[2][x]_ + Bxe[2][x] + Cx[2]e[2][x]. f) A karakterisztikus polinom _λ[n]_ _−_ 1 = _n−1_ � (λ − _e[2][πi][ k]n_ ). _k=0_ Ha n páratlan, akkor az általános megoldás _n−1_ 2 � � _y(x) = Ae[x]_ + � _Cke[cos][(][2][πi][ k]n[) ]cos e[sin][(][2][πi][ k]n[) ]+ Dke[cos][(][2][πi][ k]n[) ]sin e[sin][(][2][πi][ k]n[)]_ _,_ _k=1_ ha n páros, akkor pedig _y(x) = Ae[x]_ + Be[−][x] + _n_ 2 _[−][1]_ � _k=1_ � � _n[) ]_ _n[) ]_ _n[) ]_ _n[)]_ _Cke[cos][(][2][πi][ k]_ cos e[sin][(][2][πi][ k] + Dke[cos][(][2][πi][ k] sin e[sin][(][2][πi][ k] _._ 9. Legyenek ω0 > 0 és ω ≥ 0 valós paraméterek. Oldjuk meg az y[′′] + ω0[2][y][ = sin(][ωx][) differen-] ciálegyenletet y(0) = 0, y[′](0) = 0 kezdeti feltétel mellett. Mi történik, ha ω = ω0? _Megoldás. Az egyenlet inhomogén lineáris, a hozzá tartozó homogén egyenlet y[′′]_ + ω0[2][y][. A] karakterisztikus polinom λ[2] + ω0[2] [= (][λ][ −] _[iω][0][)(][λ][ +][ iω][0][), tehát a homogén egyenlet általános]_ megoldása y(x) = A cos(ω0x) + B sin(ω0x). Ha ω ̸= ω0, akkor nincsen külső rezonancia, az inhomogén egyenlet megoldását C cos(ωx) + D sin(ωx) alakban kereshetjük. Ezt behelyettesítve az egyenlet _−ω[2]C cos(ωx) −_ _ω[2]D sin(ωx) + ω0[2][C][ cos(][ωx][) +][ ω]0[2][D][ sin(][ωx][) = sin(][ωx][)][,]_ amiből C(ω0[2] _[−]_ _[ω][2][) = 0,][ D][(][ω]0[2]_ _[−]_ _[ω][2][) = 1. Az általános megoldás]_ 1 _y(x) =_ _ω0[2]_ _[−]_ _[ω][2][ sin(][ωx][) +][ A][ cos(][ω][0][x][) +][ B][ sin(][ω][0][x][)][.]_ Az A, B paraméterek értékét úgy kell megválasztani, hogy a kezdeti feltétel teljesüljön. _ω_ _y[′](x) =_ _ω0[2]_ _[−]_ _[ω][2][ cos(][ωx][)][ −]_ _[Aω][0][ sin(][ω][0][x][) +][ Bω][0][ cos(][ω][0][x][)]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Határozzuk meg az alábbi diﬀerenciálegyenletek általános megoldását.</span></p> <p style="top:73.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:90.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 27</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:106.4pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 25</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:122.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 18</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 81</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:139.3pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:155.7pt;left:83.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">f)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egész</span></p> <p style="top:174.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:188.7pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+10 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1+3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános megoldás</span></p> <p style="top:203.1pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:219.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 27 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános megoldás</span></p> <p style="top:234.0pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:250.5pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 25 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:264.9pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ae</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">5</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:281.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+18</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+81 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+9)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános</span></p> <p style="top:295.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Dx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:312.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az általános megoldás</span></p> <p style="top:326.7pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:343.1pt;left:83.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">f) A karakterisztikus polinom</span></p> <p style="top:376.0pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span></p> <p style="top:366.2pt;left:175.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:367.6pt;left:177.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Y</span></p> <p style="top:391.4pt;left:175.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:376.0pt;left:191.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:379.0pt;left:236.4pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:408.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> páratlan, akkor az általános megoldás</span></p> <p style="top:444.9pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:431.0pt;left:199.6pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols6,serif;font-size:6.0pt;color:#000000">−</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span></p> <p style="top:437.6pt;left:204.8pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></p> <p style="top:436.5pt;left:199.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:460.2pt;left:198.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:431.9pt;left:216.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:444.9pt;left:224.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:447.5pt;left:272.5pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:447.5pt;left:336.8pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:447.5pt;left:411.7pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:447.5pt;left:474.8pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:431.9pt;left:485.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:444.9pt;left:494.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:477.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> páros, akkor pedig</span></p> <p style="top:512.3pt;left:125.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></p> <p style="top:498.6pt;left:238.6pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i></p> <p style="top:505.0pt;left:239.1pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:503.9pt;left:239.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:527.7pt;left:238.6pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:499.3pt;left:257.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:512.3pt;left:264.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:514.9pt;left:312.8pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:514.9pt;left:377.2pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">cos</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:514.9pt;left:452.1pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">πi</span></i></sup><sup><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000"> k</span></i></sup></p> <p style="top:514.9pt;left:515.2pt;line-height:6.0pt"><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">n</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:499.3pt;left:526.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:512.3pt;left:535.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:545.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:551.9pt;left:427.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀeren-</span></sup></p> <p style="top:560.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Mi történik, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:578.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet inhomogén lineáris, a hozzá tartozó homogén egyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:584.9pt;left:509.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A</span></sup></p> <p style="top:593.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:599.3pt;left:239.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">iω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> iω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát a homogén egyenlet általános</span></sup></p> <p style="top:607.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:607.6pt;left:316.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor nincsen külső rezonan-</span></p> <p style="top:622.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cia, az inhomogén egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban kereshetjük. Ezt</span></p> <p style="top:636.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">behelyettesítve az egyenlet</span></p> <p style="top:660.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:666.9pt;left:277.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:666.9pt;left:354.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:685.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:691.2pt;left:136.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:691.2pt;left:224.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános megoldás</span></sup></p> <p style="top:714.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:706.4pt;left:162.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:722.7pt;left:145.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:728.5pt;left:152.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:744.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> paraméterek értékét úgy kell megválasztani, hogy a kezdeti feltétel teljesüljön.</span></p> <p style="top:770.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:762.7pt;left:164.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></p> <p style="top:779.0pt;left:148.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:784.8pt;left:155.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Aω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> </div>	page_299.png	8. Határozzuk meg az alábbi differenciál a á 10y-0 1) /— 124 4274 e) /— 10 4254 d) 94 1844-8ly—0 e) Y" 6y412/-—8y—0 9Y9 . Megoldás, a) A karakterisztikus polinom 324-234-10 wír) — Ae-"cos3z 4 Be-"sin3z. 1) A karakterisztikus polinom 3? — 129 427 — () — 3)(A — 9), az általános megoldás víz) — A" 4 B. e) A karakterisztikus polinom 3? — 10X 4. 25 — () — 5), az általános megoldás yíz) — Aéz 4 Bre. d) A karakterisztikus polinom A! 4-18324-81 — (24-9)? — (14-372(4—31), az általános megoldás yíz) — Acos3r -- Bsinőr 4 Crcos3z 4 Dzsi e) A karakterisztikus polinom X? — 632 4- 124 — 8 wíz) — Aett 4 Bzett 4 CrPé, £) A karakterisztikus polinom aletek általános megoldását. 0. ahol n 2 1 egész A41—30)(14-1--38), az általános megoldás "l;[U é F Ha n páratlan, akkor az általános megoldás aY (mw—v £ 4 DuElESE) sin t A--*:)) ha n páros, akkor pedig k ] 9. Legyenek aa 2. 0 és az 20 valós paraméterek. Oldjuk meg az 4" 4 ugy — sin(oz) díflerer ciálegyenletet y(0) — 0, 4(0) — 0 kezdeti feltétel mellett. Mi történik, ha 4 — 9? Megoldás. Az egy megoldása v(z) " alet inhomo 3 —(X — íaaJ(X 4-áuv), tehát a homogén egyenlet általános Atoslzasz) t Bsin(uoz). Ha 4 28 , akkor ninc ) 4. Dsin külső rezonai ) alakban kereshetjűk. Ezt behelyettesítve az egyen ?C cos(cor) — sZDsin(r) 4 aC cos(ez) 4 sgDsin(or sin(en), amiből C( — 2) — 0, Díu — a) — 1. Az általános megoldás v sánssz) 4 Acoslanz) 4 Bsintonz) úgy kell megválasztani, hogy a kezdeti fel os(coz) — Azoysinleoyz) 4. Buoy coslaz) 6
felhasználásával a feltétel 0 = y(0) = A 0 = y′(0) = ω ω2 0 −ω2 + Bω0, tehát A = 0 és B = − ω ω0(ω2 0−ω2), a kezdetiérték-probléma megoldása y(x) = ω2 0 1 ω2 0 0 −ω2 sin(ωx) − ω ω0(ω2 ω ω2 ω0(ω2 0 −ω2) sin(ω0x) = ω0 sin(ωx) −ω sin(ω0x) 0 −ω2) ω0(ω2 0 ω2 0 −ω2) Ha ω = ω0, akkor külső rezonancia van, tehát az inhomogén egyenletnek Cx cos(ω0x) + Dx sin(ω0x) alakban keressük a megoldását. Behelyettesítés után az egyenlet 2Dω0 cos(ω0x) −2Cω0 sin(ω0x) = sin(ω0x), tehát C = −1 2ω0 és D = 0. Az általános megoldás y(x) = −1 2ω0 x cos(ω0x) + A cos(ω0x) + B sin(ω0x). A kezdeti feltételből 0 = y(0) = A 0 = y′(0) = −1 2ω0 + Bω0, azaz A = 0 és B = 1 2ω2 0 . A kezdetiérték-probléma megoldása ekkor y(x) = −1 2ω0 x cos(ω0x) + 1 2ω2 0 sin(ω0x). Tehát külső rezonancia esetén a megoldás nem lesz korlátos, az első tag miatt lineárisan nő a rezgés amplitúdója. Ezt a megoldást az ω ̸= ω0 melletti megoldásból ω →ω0 határátmenettel is megkaphattuk volna. 10. Legyenek ω1, ω2 ≥0 valós paraméterek. Oldjuk meg az y(4) + (ω2 2)y′′ + ω2 1ω2 2y = 0 1 + ω2 ω2 2 1 + ω2 ω2 2)y′′ + ω2 1 ω2 1ω2 2 Legyenek ω1, ω2 ≥0 valós paraméterek. Oldjuk meg az y(4) + (ω2 2)y′′ + ω2 1ω2 1 + ω2 2y = 0 diﬀerenciálegyenletet y(0) = 1, y′(0) = y′′(0) = y′′′(0) = 0 kezdeti feltétel mellett. Mi történik, ha ω1 = ω2? Megoldás. Az egyenlet homogén lineáris állandó együtthatós, a karakterisztikus polinom λ4 + (ω2 2)λ2 + ω2 1ω2 1)(λ2 + ω2 2). Ha ω1 ̸= ω2, akkor nincs rezonancia, az 1 + ω2 2 = (λ2 + ω2 2 2 1 + ω2 z egyenlet ω2 2)λ2 + ω2 1 ω2 1ω2 2 homogén lineá ω2 1 2 = (λ2 + ω2 eáris álland ω2 1)(λ2 + ω2 2 λ4 + (ω2 2)λ2 + ω2 1ω2 1)(λ2 + ω2 1 + ω2 2 = (λ2 + ω2 2). Ha ω1 ̸= ω2, akkor nincs rezonancia, az általános megoldás y(x) = A cos ω1x + B sin ω1x + C cos ω1x + D sin ω2x. Ennek deriváltjai: y′(x) = Bω1 cos ω1 −Aω1 sin ω1x + Dω2 cos ω1x −Cω1 sin ω1x y′′(x) = −Aω2 1 cos ω1 −Bω2 1 sin ω1x −Cω2 2 cos ω1x −Dω2 1 sin ω ω2 1 1 cos ω1 −Bω2 2 2 1 sin ω1x −Cω2 ω2 1 2 cos ω1x −Dω2 y′′(x) = −Aω2 1 cos ω1 −Bω2 1 sin ω1x −Cω2 2 cos ω1x −Dω2 1 sin ω1x y′′′(x) = −Bω3 1 cos ω1 + Aω3 1 sin ω1x −Dω3 2 cos ω1x + Cω3 1 sin ω1x. ω3 1 1 cos ω1 + Aω3 ω3 2 1 sin ω1x −Dω3 1 3 2 cos ω1x + Cω3 ω3 1 sin ω1x. A kezdeti feltétel alapján az együtthatókra az 1 = y(0) = A + C 0 = y′(0) = ω1B + ω2D 0 = y′′(0) = −ω2 1A −ω2 2 ω2 1A −ω2 2 0 = y′′(0) = −ω2 1A −ω2 2C 0 = y′′′(0) = −ω3 1B −ω3 2D ω3 1B −ω3 2 ω3 2D	felhasználásával a feltétel 0 = y(0) = A _ω_ 0 = y[′](0) = _ω0[2]_ _[−]_ _[ω][2][ +][ Bω][0][,]_ tehát A = 0 és B = − _ω_ _ω0(ω0[2][−][ω][2][)]_ [, a kezdetiérték-probléma megoldása] 1 _ω_ _y(x) =_ _._ _ω0[2]_ _[−]_ _[ω][2][ sin(][ωx][)][ −]_ _ω0(ω0[2]_ _[−]_ _[ω][2][) sin(][ω][0][x][) =][ ω][0][ sin(]ω[ωx]0(ω[)][ −]0[2]_ _[−][ω][ω][ sin(][2][)]_ _[ω][0][x][)]_ Ha ω = ω0, akkor külső rezonancia van, tehát az inhomogén egyenletnek Cx cos(ω0x) + _Dx sin(ω0x) alakban keressük a megoldását. Behelyettesítés után az egyenlet_ 2Dω0 cos(ω0x) − 2Cω0 sin(ω0x) = sin(ω0x), tehát C = − [1] 2ω0 [és][ D][ = 0. Az általános megoldás] _y(x) = −_ [1] _x cos(ω0x) + A cos(ω0x) + B sin(ω0x)._ 2ω0 A kezdeti feltételből 0 = y(0) = A 0 = y[′](0) = − [1] + Bω0, 2ω0 azaz A = 0 és B = 1 2ω0[2] [. A kezdetiérték-probléma megoldása ekkor] 1 _y(x) = −_ [1] _x cos(ω0x) +_ sin(ω0x). 2ω0 2ω0[2] Tehát külső rezonancia esetén a megoldás nem lesz korlátos, az első tag miatt lineárisan nő a rezgés amplitúdója. Ezt a megoldást az ω ̸= ω0 melletti megoldásból ω → _ω0 határátme-_ nettel is megkaphattuk volna. 10. Legyenek ω1, ω2 ≥ 0 valós paraméterek. Oldjuk meg az y[(4)] + (ω1[2] [+][ ω]2[2][)][y][′′][ +][ ω]1[2][ω]2[2][y][ = 0] differenciálegyenletet y(0) = 1, y[′](0) = y[′′](0) = y[′′′](0) = 0 kezdeti feltétel mellett. Mi történik, ha ω1 = ω2? _Megoldás. Az egyenlet homogén lineáris állandó együtthatós, a karakterisztikus polinom_ _λ[4]_ + (ω1[2] [+][ ω]2[2][)][λ][2][ +][ ω]1[2][ω]2[2] [= (][λ][2][ +][ ω]1[2][)(][λ][2][ +][ ω]2[2][). Ha][ ω][1] _[̸][=][ ω][2][, akkor nincs rezonancia, az]_ általános megoldás _y(x) = A cos ω1x + B sin ω1x + C cos ω1x + D sin ω2x._ Ennek deriváltjai: _y[′](x) = Bω1 cos ω1 −_ _Aω1 sin ω1x + Dω2 cos ω1x −_ _Cω1 sin ω1x_ _y[′′](x) = −Aω1[2]_ [cos][ ω][1] _[−]_ _[Bω]1[2]_ [sin][ ω][1][x][ −] _[Cω]2[2]_ [cos][ ω][1][x][ −] _[Dω]1[2]_ [sin][ ω][1][x] _y[′′′](x) = −Bω1[3]_ [cos][ ω][1] [+][ Aω]1[3] [sin][ ω][1][x][ −] _[Dω]2[3]_ [cos][ ω][1][x][ +][ Cω]1[3] [sin][ ω][1][x.] A kezdeti feltétel alapján az együtthatókra az 1 = y(0) = A + C 0 = y[′](0) = ω1B + ω2D 0 = y[′′](0) = −ω1[2][A][ −] _[ω]2[2][C]_ 0 = y[′′′](0) = −ω1[3][B][ −] _[ω]2[3][D]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felhasználásával a feltétel</span></p> <p style="top:81.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:103.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></p> <p style="top:95.3pt;left:184.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></p> <p style="top:111.6pt;left:168.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:117.4pt;left:176.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:131.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:130.2pt;left:209.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i></p> <p style="top:139.1pt;left:191.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:143.4pt;left:209.5pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a kezdetiérték-probléma megoldása</span></sup></p> <p style="top:165.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:156.9pt;left:162.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:173.2pt;left:145.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:179.0pt;left:152.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:156.9pt;left:268.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></p> <p style="top:173.2pt;left:241.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:179.0pt;left:265.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:173.2pt;left:393.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:179.0pt;left:417.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:165.0pt;left:484.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:194.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor külső rezonancia van, tehát az inhomogén egyenletnek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></p> <p style="top:208.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Dx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban keressük a megoldását. Behelyettesítés után az egyenlet</span></p> <p style="top:231.6pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Dω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Cω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:254.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:261.7pt;left:143.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános megoldás</span></sup></p> <p style="top:284.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:292.7pt;left:154.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:284.5pt;left:173.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:312.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltételből</span></p> <p style="top:334.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:358.7pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:366.9pt;left:178.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:358.7pt;left:199.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:387.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:386.3pt;left:182.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:395.2pt;left:177.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:399.5pt;left:187.1pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A kezdetiérték-probléma megoldása ekkor</span></sup></p> <p style="top:419.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:427.9pt;left:154.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:419.7pt;left:173.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></p> <p style="top:411.6pt;left:247.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:427.9pt;left:241.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:433.7pt;left:254.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:419.7pt;left:262.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:448.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Tehát külső rezonancia esetén a megoldás nem lesz korlátos, az első tag miatt lineárisan nő</span></p> <p style="top:462.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a rezgés amplitúdója. Ezt a megoldást az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> melletti megoldásból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> határátme-</span></p> <p style="top:476.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nettel is megkaphattuk volna.</span></p> <p style="top:494.8pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós paraméterek. Oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:500.9pt;left:418.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:500.9pt;left:446.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:500.9pt;left:490.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:500.9pt;left:502.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup></p> <p style="top:509.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett. Mi</span></p> <p style="top:523.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">történik, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">?</span></p> <p style="top:541.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet homogén lineáris állandó együtthatós, a karakterisztikus polinom</span></p> <p style="top:556.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.1pt;left:116.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.1pt;left:144.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.1pt;left:188.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.1pt;left:200.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.1pt;left:262.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.1pt;left:311.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ha</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"≯</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor nincs rezonancia, az</span></sup></p> <p style="top:570.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldás</span></p> <p style="top:593.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></p> <p style="top:615.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ennek deriváltjai:</span></p> <p style="top:638.6pt;left:111.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Aω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Dω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Cω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:656.1pt;left:108.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Aω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:662.2pt;left:176.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Bω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:662.2pt;left:245.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Cω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:662.2pt;left:318.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Dω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:662.2pt;left:393.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:673.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Bω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:679.7pt;left:177.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Aω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:679.7pt;left:244.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Dω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:679.7pt;left:319.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:679.7pt;left:393.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></sup></p> <p style="top:696.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltétel alapján az együtthatókra az</span></p> <p style="top:719.0pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></p> <p style="top:736.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></p> <p style="top:753.9pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:760.0pt;left:186.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:760.0pt;left:222.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup></p> <p style="top:771.3pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:777.4pt;left:188.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:777.4pt;left:225.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> </div>	page_300.png	felhasználásával a feltétel 0-v(0) Oa é t Ba, v - p po té - É Ha 1 — , akkor külső rezonancia van, tehát az inhomogén egyenletnek €- cos(4uz) 4 Drsin(cooz) alakban keressük a megoldását. Behelyettesítés után az egyenlet 2D coslcayr) — 2Csay sinlosz) — sinlosr), k és D —0. Az általános n tehát €-— —h egoldás FDE Lihum(_u,:) 4 Acos(zoy) t Bsin(suyr). A kezdeti felt 1 4 Buz, FDE Lihum(_u,:) 4 zNiá_,..(wl) Tehát külső rezonancia esetén a megoldás nem lesz korlátos, az első tag miatt lincárisai a rezgés amplitúdója. Ezt a megoldást az ss 2 3) melletti megoldásból c; — 44 határátme- ettel ís megkaphattuk volna. Legyenek az,sös 2 0 valós paraméterek. Oldjuk meg az y/9 4- (s? 4 sZyy" 4 fely — 0 erenciálegyenletet y(0) — 1. y(0) — y"(0) — w"(0) — 0 kezdeti feltétel mellett. Mi X84h 48 4 aa — O? 4-JO? 4489. Ha úx 2 son, akkor nincs rezonancia, az wíz) — Acoszar 4 Bsincar 4 Ccosiazr 4 Dsinusz. W(G) — B cos a — Azysineo, víz A kezdeti feltét 4 Diez coscar — Casiner Azol cos a — B ss — Csol eossn — Decl sit Basj cosaz 4 Azol sin De cosenz 4 Css sit tel alapján az együtthatókra az
egyenletrendszer teljesül, ennek megoldása A = ω2 2 ω2 ω2 2 ω2 1 2 −ω2 B = 0 C = − ω2 1 2 ω2 2 1 2 2 −ω2 D = 0, tehát a kezdetiérték-probléma megoldása 2 y(x) = ω2 ω2 1 2 cos ω1x −ω2 ω1x −ω2 1 cos ω2x ω2 1 2 −ω2 ω2 1 2 −ω2 Ha ω1 = ω2 =: ω, akkor belső rezonancia van, tehát az általános megoldás y(x) = A cos ωx + B sin ωx + Cx cos ωx + Dx sin ωx, ebből hasonlóan azt kapjuk, hogy y(x) = 1 2xω sin ωx + cos ωx. Ez a megoldás egyébként azzal a függvénnyel is megegyezik, amit az ω1 ̸= ω2 esetben kapottból ω2 →ω1 határátmenettel kapunk. 11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását. g p g a) y′′ + 4y′ + 8y = e−2x cos 2x, y(0) = 1, y′(0) = 0 b) y′′′ −3y′′ −y′ + 3y = xe2x, y(0) = y′(0) = y′′(0) = 0 ) y y y y , y( ) y ( ) y ( ) c) y′′′ −3y′′ −y′ + 3y = xe−x, y(0) = y′(0) = y′′(0) = 0 d) y′′ + 8y′ + 16y = x2e−4x, y(0) = 0, y′(0) = 1 Megoldás. g a) A karakterisztikus polinom λ2 + 4λ + 8 = (λ + 2 + 2i)(λ + 2 −2i), a homogén egyenlet A karakterisztikus polinom λ + 4λ + 8 (λ + 2 + 2i)(λ + 2 2i), a homogén egyenlet általános megoldása y(x) = Ae−2x cos 2x+Be−2x sin 2x. Külső rezonancia van, az inhomogén egyenlet egy megoldását y(x) = Cxe−2x cos 2x+Dxe−2x sin 2x alakban keressük, behelyettesítés után C = 0, D = 1 4 adódik. Az általános megoldás és deriváltja 1 4 adódik. Az általános megoldás és deriváltja y(x) = 1 4 1 4xe−2x sin 2x + Ae−2x cos 2x + Be−2x sin 2x y′(x) = 1 4 1 4e−2x sin 2x −1 2 1 2xe−2x sin 2x + 1 2 1 2xe−2x cos 2x 4 2 2 + (−2A + 2B)e−2x cos 2x + (−2A −2B)e−2x sin 2x, a kezdeti feltétel alapján 1 = y(0) = A 0 = y′(0) = −2A + 2B, tehát A = B = 1. b) A karakterisztikus polinom λ3 −3λ2 −λ+3 = (λ+1)(λ−1)(λ−3), a homogén egyenlet általános megoldása y(x) = Ae−x + Bex + Ce3x. Nincsen rezonancia, az inhomogén egyenlet megoldását y(x) = (C0 + C1x)e2x alakban keressük. Behelyettesítés után az	egyenletrendszer teljesül, ennek megoldása _A =_ _ω2[2]_ _ω2[2]_ _[−]_ _[ω]1[2]_ _B = 0_ _C = −_ _ω1[2]_ _ω2[2]_ _[−]_ _[ω]1[2]_ _D = 0,_ tehát a kezdetiérték-probléma megoldása 2 [cos][ ω][1][x][ −] _[ω]1[2]_ [cos][ ω][2][x] _y(x) =_ _[ω][2]_ _._ _ω2[2]_ _[−]_ _[ω]1[2]_ Ha ω1 = ω2 =: ω, akkor belső rezonancia van, tehát az általános megoldás _y(x) = A cos ωx + B sin ωx + Cx cos ωx + Dx sin ωx,_ ebből hasonlóan azt kapjuk, hogy _y(x) = [1]_ 2[xω][ sin][ ωx][ + cos][ ωx.] Ez a megoldás egyébként azzal a függvénnyel is megegyezik, amit az ω1 ̸= ω2 esetben kapottból ω2 → _ω1 határátmenettel kapunk._ 11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását. a) y[′′] + 4y[′] + 8y = e[−][2][x] cos 2x, y(0) = 1, y[′](0) = 0 b) y[′′′] _−_ 3y[′′] _−_ _y[′]_ + 3y = xe[2][x], y(0) = y[′](0) = y[′′](0) = 0 c) y[′′′] _−_ 3y[′′] _−_ _y[′]_ + 3y = xe[−][x], y(0) = y[′](0) = y[′′](0) = 0 d) y[′′] + 8y[′] + 16y = x[2]e[−][4][x], y(0) = 0, y[′](0) = 1 _Megoldás._ a) A karakterisztikus polinom λ[2] + 4λ + 8 = (λ + 2 + 2i)(λ + 2 − 2i), a homogén egyenlet általános megoldása y(x) = Ae[−][2][x] cos 2x + _Be[−][2][x]_ sin 2x. Külső rezonancia van, az inhomogén egyenlet egy megoldását y(x) = Cxe[−][2][x] cos 2x+Dxe[−][2][x] sin 2x alakban keressük, behelyettesítés után C = 0, D = [1] 4 [adódik. Az általános megoldás és deriváltja] _y(x) = [1]_ 4[xe][−][2][x][ sin 2][x][ +][ Ae][−][2][x][ cos 2][x][ +][ Be][−][2][x][ sin 2][x] _y[′](x) = [1]_ 4[e][−][2][x][ sin 2][x][ −] 2[1][xe][−][2][x][ sin 2][x][ + 1]2[xe][−][2][x][ cos 2][x] + (−2A + 2B)e[−][2][x] cos 2x + (−2A − 2B)e[−][2][x] sin 2x, a kezdeti feltétel alapján 1 = y(0) = A 0 = y[′](0) = −2A + 2B, tehát A = B = 1. b) A karakterisztikus polinom λ[3] _−_ 3λ[2] _−_ _λ_ +3 = (λ +1)(λ _−_ 1)(λ _−_ 3), a homogén egyenlet általános megoldása y(x) = Ae[−][x] + Be[x] + Ce[3][x]. Nincsen rezonancia, az inhomogén egyenlet megoldását y(x) = (C0 + C1x)e[2][x] alakban keressük. Behelyettesítés után az -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszer teljesül, ennek megoldása</span></p> <p style="top:91.9pt;left:107.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:83.8pt;left:147.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:90.0pt;left:154.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:100.1pt;left:133.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:105.9pt;left:140.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:105.9pt;left:167.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:116.8pt;left:107.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:142.7pt;left:107.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:134.6pt;left:156.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:140.8pt;left:163.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:150.9pt;left:142.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:156.7pt;left:150.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:156.7pt;left:177.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:167.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:194.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a kezdetiérték-probléma megoldása</span></p> <p style="top:226.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:224.9pt;left:152.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:224.9pt;left:218.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:235.1pt;left:183.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:240.9pt;left:190.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:240.9pt;left:218.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:226.9pt;left:262.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:258.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =:</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor belső rezonancia van, tehát az általános megoldás</span></p> <p style="top:285.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ωx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Dx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ωx,</span></i></p> <p style="top:311.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ebből hasonlóan azt kapjuk, hogy</span></p> <p style="top:342.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:350.8pt;left:145.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xω</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ωx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ωx.</span></i></sup></p> <p style="top:372.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez a megoldás egyébként azzal a függvénnyel is megegyezik, amit az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetben</span></p> <p style="top:386.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kapottból</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> határátmenettel kapunk.</span></p> <p style="top:405.9pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11. Határozzuk meg az alábbi kezdetiérték-problémák megoldását.</span></p> <p style="top:420.3pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span></p> <p style="top:436.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span></p> <p style="top:453.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span></p> <p style="top:469.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span></p> <p style="top:489.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:503.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 8 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet</span></p> <p style="top:518.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Be</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Külső rezonancia van, az inho-</span></p> <p style="top:532.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mogén egyenlet egy megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cxe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Dxe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakban keressük,</span></p> <p style="top:546.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">behelyettesítés után</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:554.2pt;left:268.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">adódik. Az általános megoldás és deriváltja</span></sup></p> <p style="top:579.5pt;left:129.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:587.7pt;left:168.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:607.3pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:615.5pt;left:168.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:615.5pt;left:243.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:615.5pt;left:325.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:629.6pt;left:154.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x,</span></i></p> <p style="top:656.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a kezdeti feltétel alapján</span></p> <p style="top:682.4pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:699.9pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">B,</span></i></p> <p style="top:726.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:742.7pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet</span></p> <p style="top:757.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Nincsen rezonancia, az inhomogén</span></p> <p style="top:771.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük. Behelyettesítés után az</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> </div>	page_301.png	egyenletrendszer teljesül, ennek megoldása tehát a kezdetiérték-probléma megoldása v Fi wíz) — Acoscor 4 Bsincor 4 Crcoscor 4 Drsin ebből hasonlóan azt kapjuk, hogy 1 EŐ] kapottból azz — 221 határátmenettel kapunk. a) Y 4444 8y — €-7 cos2r, v(0) — 1. V(0) 0 1) Y" gyY — 43 rén y0) - y0)-Y(0-0 e) — 347 — Y 4.3y — ze7", y(0) — (0) — Y(0) d) 7 4-.8y/ 4 16y — 2é-. y(0) — 0, 7(0) Megoldás. a) A karakterisztikus polinom X? 44X 4-8 általános megoldása y(z mogén egyenlet egy megoldását y(z) — Cxe-?" cos2z-4-Dre-?" sin 27 alakban keressük, behelyettesítés után C — 0. D — 1 adódik. Az általános megoldás és deriváltja egegyezik, amit az tx 2 ú2 csetben O4-24-20)(44-2—21), a homogén egye ex cos2z- Be "" sin 2r. Külső rezonancia van, az inh. v ! " 162 1o0-2s, e sin2r — gze t sin2r 4 zze t cos2r 1 : 7 (-2A4-2B)e-"tcos2z 4 (-2A — 2BJe-?" én. a kezdeti felté el alapján. 2A 2B. tehát A B—1. A karakterisztikus polinom 39—332—34-3 — (14-1—1)(4—3), a homogén egye általános megoldása y(r) egyenlet megoldását yíz) 1) let A" 4 Be 4. Ce?", Nincsen rezonancia, az inhomogó (Co 4 Ciz)e?: alakban keressük. Behelyettesítés után az
egyenlet (−3C0 −C1 −3C1x)e2x = xe2x, amiből C0 = 1 9 3 1 9 és C1 = −1 egyenlet (−3C0 −C1 −3C1x)e2x = xe2x, amiből C0 = 1 9 és C1 = −1 3. Az általános megoldás és deriváltjai y(x) = 1 9 1 9e2x −1 3 1 3xe2x + Ae−x + Bex + Ce3x y′(x) = −1 9 1 9e2x −2 3 2 3xe2x −Ae−x + Bex + 3Ce3x y′′(x) = −8 9 8 9e2x −4 3 4 3xe2x + Ae−x + Bex + 9Ce3x, a kezdeti feltétel alapján 0 = y(0) = 1 9 1 9 + A + B + C 0 = y′(0) = −1 9 1 9 −A + B + 3C 0 = y′′(0) = −8 9 8 9 + A + B + 9C, ennek megoldása A = 1 72, B = −1 4 1 4, C = 1 8 ennek megoldása A = 1 72, B = −1 4, C = 1 8. c) A karakterisztikus polinom λ3 −3λ2 −λ+3 = (λ+1)(λ−1)(λ−3), a homogén egyenlet p + ( + )( )( ), g gy általános megoldása y(x) = Ae−x + Bex + Ce3x. Külső rezonancia van, az inhomogén egyenlet megoldását y(x) = (C0 + C1x)xe−x alakban keressük. Behelyettesítés után az egyenlet (8C0 −12C1 + 16C1x)e−x = xe−x, amiből C0 = 3 32 és C1 = 1 16. Az általános megoldás és deriváltjai y(x) = 3 32 3 32xe−x + 1 16 1 16x2e−x + Ae−x + Bex + Ce3x y′(x) = 3 32 3 32e−x + 1 32 1 32xe−x −1 16 1 16x2e−x −Ae−x + Bex + 3Ce3x y′′(x) = −1 16 1 16e−x −5 32 5 32xe−x + 1 16 1 16x2e−x + Ae−x + Bex + 9Ce3x, a kezdeti feltétel alapján 0 = y(0) = A + B + C 0 = y′(0) = 3 32 3 32 −A + B + 3C 0 = y′′(0) = −1 16 1 16 + A + B + 9C, ennek megoldása A = 7 128, B = −1 16 ennek megoldása A = 7 128, B = −1 16, C = 1 128. d) A karakterisztikus polinom λ2 + 8λ + 16 = (λ + 4)2, a homogén egyenlet általános a a te s t us po o λ + 8λ + 6 (λ + ) , a o ogé egye et á ta á os megoldása y(x) = Ae−4x + Bxe−4x (belső rezonancia). Külső rezonancia is van, az inhomogén egyenlet megoldását y(x) = (C0 + C1x + C2x2)x2e−4x alakban keressük. Behelyettesítve a kapott egyenlet (2C0 + 6C1x + 12C2x2)e−4x = x2e−4x, tehát C0 = C1 = 0, C2 = 1 12. Az általános megoldás és deriváltja y(x) = 1 12 1 12x4e−4x + Ae−4x + Bxe−4x y′(x) = 1 3 1 3x3e−4x −1 3 1 3x4e−3x + (−4A + B)e−4x −4Bxe−4x, a kezdeti feltétel alapján 0 = y(0) = A 1 = y′(0) = −4A + B, ennek megoldása A = 0, B = 1.	egyenlet (−3C0 − _C1 −_ 3C1x)e[2][x] = xe[2][x], amiből C0 = [1]9 [és][ C][1][ =][ −] [1]3 [. Az általános] megoldás és deriváltjai _y(x) = [1]_ 9[e][2][x][ −] [1]3[xe][2][x][ +][ Ae][−][x][ +][ Be][x][ +][ Ce][3][x] _y[′](x) = −[1]_ 9[e][2][x][ −] 3[2][xe][2][x][ −] _[Ae][−][x][ +][ Be][x][ + 3][Ce][3][x]_ _y[′′](x) = −[8]_ 9[e][2][x][ −] 3[4][xe][2][x][ +][ Ae][−][x][ +][ Be][x][ + 9][Ce][3][x][,] a kezdeti feltétel alapján 0 = y(0) = [1] 9 [+][ A][ +][ B][ +][ C] 0 = y[′](0) = −[1]9 _[−]_ _[A][ +][ B][ + 3][C]_ 0 = y[′′](0) = −[8] 9 [+][ A][ +][ B][ + 9][C,] ennek megoldása A = 1 72 [,][ B][ =][ −] [1]4 [,][ C][ =][ 1]8 [.] c) A karakterisztikus polinom λ[3] _−_ 3λ[2] _−_ _λ_ +3 = (λ +1)(λ _−_ 1)(λ _−_ 3), a homogén egyenlet általános megoldása y(x) = Ae[−][x] + Be[x] + Ce[3][x]. Külső rezonancia van, az inhomogén egyenlet megoldását y(x) = (C0 + C1x)xe[−][x] alakban keressük. Behelyettesítés után az egyenlet (8C0 − 12C1 + 16C1x)e[−][x] = xe[−][x], amiből C0 = 323 [és][ C][1][ =] 161 [. Az általános] megoldás és deriváltjai _y(x) = [3]_ 32[xe][−][x][ + 1]16[x][2][e][−][x][ +][ Ae][−][x][ +][ Be][x][ +][ Ce][3][x] _y[′](x) = [3]_ 32[e][−][x][ + 1]32[xe][−][x][ −] 16[1] _[x][2][e][−][x][ −]_ _[Ae][−][x][ +][ Be][x][ + 3][Ce][3][x]_ _y[′′](x) = −_ [1] 16[e][−][x][ −] 32[5] _[xe][−][x][ + 1]16[x][2][e][−][x][ +][ Ae][−][x][ +][ Be][x][ + 9][Ce][3][x][,]_ a kezdeti feltétel alapján 0 = y(0) = A + B + C 0 = y[′](0) = 32[3] _[−]_ _[A][ +][ B][ + 3][C]_ 0 = y[′′](0) = − [1] 16 [+][ A][ +][ B][ + 9][C,] ennek megoldása A = 7 1 128 [,][ B][ =][ −] 16[1] [,][ C][ =] 128 [.] d) A karakterisztikus polinom λ[2] + 8λ + 16 = (λ + 4)[2], a homogén egyenlet általános megoldása y(x) = Ae[−][4][x] + Bxe[−][4][x] (belső rezonancia). Külső rezonancia is van, az inhomogén egyenlet megoldását y(x) = (C0 + C1x + C2x[2])x[2]e[−][4][x] alakban keressük. Behelyettesítve a kapott egyenlet (2C0 + 6C1x + 12C2x[2])e[−][4][x] = x[2]e[−][4][x], tehát C0 = _C1 = 0, C2 =_ 121 [. Az általános megoldás és deriváltja] _y(x) = [1]_ 12[x][4][e][−][4][x][ +][ Ae][−][4][x][ +][ Bxe][−][4][x] _y[′](x) = [1]_ 3[x][3][e][−][4][x][ −] [1]3[x][4][e][−][3][x][ + (][−][4][A][ +][ B][)][e][−][4][x][ −] [4][Bxe][−][4][x][,] a kezdeti feltétel alapján 0 = y(0) = A 1 = y[′](0) = −4A + B, ennek megoldása A = 0, B = 1. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:66.4pt;left:389.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">9</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:66.4pt;left:457.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános</span></sup></p> <p style="top:73.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás és deriváltjai</span></p> <p style="top:99.0pt;left:132.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:107.2pt;left:170.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:107.2pt;left:208.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:126.8pt;left:129.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:135.0pt;left:180.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:135.0pt;left:217.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:154.6pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></sup></p> <p style="top:162.8pt;left:180.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup></p> <p style="top:162.8pt;left:217.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:178.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a kezdeti feltétel alapján</span></p> <p style="top:204.1pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:212.3pt;left:186.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup></p> <p style="top:231.9pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:240.1pt;left:198.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup></p> <p style="top:259.7pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></sup></p> <p style="top:267.9pt;left:201.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C,</span></i></sup></p> <p style="top:285.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:283.8pt;left:215.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:292.6pt;left:213.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">72</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:292.6pt;left:266.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 1</span></sup></p> <p style="top:292.6pt;left:304.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">8</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:301.7pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+3 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet</span></p> <p style="top:316.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Külső rezonancia van, az inhomogén</span></p> <p style="top:330.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük. Behelyettesítés után az</span></p> <p style="top:345.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 16</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:343.5pt;left:397.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:352.4pt;left:395.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">32</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:343.5pt;left:457.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:352.4pt;left:455.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">16</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános</span></sup></p> <p style="top:359.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás és deriváltjai</span></p> <p style="top:385.0pt;left:132.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup></p> <p style="top:393.2pt;left:170.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:393.2pt;left:223.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:412.8pt;left:129.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup></p> <p style="top:421.0pt;left:170.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:421.0pt;left:216.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:421.0pt;left:269.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:440.6pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:448.8pt;left:180.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></sup></p> <p style="top:448.8pt;left:226.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:448.8pt;left:278.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:464.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a kezdeti feltétel alapján</span></p> <p style="top:485.6pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></p> <p style="top:509.5pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup></p> <p style="top:517.7pt;left:189.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">32</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></sup></p> <p style="top:537.3pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:545.5pt;left:201.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">16 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C,</span></i></sup></p> <p style="top:562.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:561.4pt;left:217.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">7</span></p> <p style="top:570.2pt;left:213.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">128</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:570.2pt;left:270.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">16</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:561.4pt;left:317.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:570.2pt;left:313.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">128</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:579.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) A karakterisztikus polinom</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 16 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a homogén egyenlet általános</span></p> <p style="top:593.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(belső rezonancia). Külső rezonancia is van, az</span></p> <p style="top:608.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">inhomogén egyenlet megoldását</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keressük.</span></p> <p style="top:622.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Behelyettesítve a kapott egyenlet</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:637.1pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:635.6pt;left:171.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:644.4pt;left:169.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">12</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az általános megoldás és deriváltja</span></sup></p> <p style="top:664.4pt;left:129.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:672.6pt;left:168.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bxe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:692.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:700.4pt;left:168.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:700.4pt;left:224.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Bxe</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:716.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a kezdeti feltétel alapján</span></p> <p style="top:737.2pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:754.6pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B,</span></i></p> <p style="top:775.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> </div>	page_302.png	egyenlet (-3Cy9 — C — 3CE megoldás és deriváltjai zét", amiből C — § és C — —1. Az általános É AT A BE CT 1 e) F ér — AT 4 BE 4 307 v -- ! s v - -5e ét 4 A 4 BE 4. 9C€", a kezdeti feltétel alapján 1 j7asBsC ö-vi -l ba90 po - -8 7 0-v(0)--§4448 490 ennek megoldása A — $. B -- C - A karakterisztikus polinom 99—332—94-3 — (14-1(A-—1)()—3), a homogén egyenlet egyenlet megoldását yíz) — (Cy 4 Ciz)ze"" alakban keressük. B. egyenlet (8Cy — 12C, 4 16C12)e amiből C — § és C Az általános vACTABÉGCÉT te ét á Be 430" 16 ZE 4 ggrlE T 4 AGT 4 BE 490 a kezdeti feltétel alapján. AZBAC AZBABC ennek megoldása A B 0 í A karakterisztikus polinom 9? 4. 8X 4. 16 — (X 4-4)2, a homogén egyenlet általános inhomogén egyenlet megoldását yíz) — (Ca 4 Cir 4 Csr?)?e-2 alakban keressük. Behelyettesítve a kapott egyenlet (2Cy 4601x 4 1204z2je- — r2e-tt, tehát C — C1—0. Ca — £3. Az általános megoldás és deriváltja 1t , A m vle) 2 gzeté 4 AE" 4 Bze v ét te a (-444 Byet ar t. a kezdeti feltétel alapján 0-v0)-A 1-v(0)-—444 5. s megoldása A — 0, B— 1.
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 11. feladatsor: Laplace-transzformáció 1. Határozzuk meg az f : (0, ∞) →R függvény Laplace-transzformáltját. a) f(x) = cos2 x b)   0 ha x < a x−a b−a ha a ≤x < b 1 ha x ≥b f(x) = ahol 0 < a < b. 2. Határozzuk meg Laplace-transzformációval az y′′′+y = 1 diﬀerenciálegyenlet y(0) = y′(0) = y′′(0) = 0 kezdeti feltételt kielégítő megoldását. 3. Laplace-transzformáció alkalmazásával oldjuk meg az xy′′ + 2y′ + xy = 0 diﬀerenciálegyen- letet. 4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze. Az egyiket a t0 = 0 pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük. Ha y1, y2 jelöli az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó diﬀerenciálegyenlet-rendszer y′′ 1 y′′ 1 = y2 −y1 + f y′′ 2 = y1 −y2. y′′ 2 = y1 −y2. Hogyan mozognak a t0 = 0 pillanat után? További gyakorló feladatok 5. Határozzuk meg az f : (0, ∞) →R függvény Laplace-transzformáltját. a) f(x) = x−1/2 b) f(x) = sgn sin(πx) 6. Laplace-transzformáció segítségével határozzuk meg az y′′ −2y′ +y = x diﬀerenciálegyenlet y(0) = 0 y′(0) = −1 kezdeti feltételt kielégítő megoldását. 7. Laplace-transzformáció segítségével oldjuk meg az y′′ + 2y′ + 2y = 0 diﬀerenciálegyenletet y(0) = 0, y′(0) = 1 kezdeti feltétel mellett. 8. Laplace-transzformáció alkalmazásával oldjuk meg az xy′′ + 2y′ −xy = 0 diﬀerenciálegyen- letet. 9. Laplace-transzformáció alkalmazásával oldjuk meg az y′ 1 y′ 1 = −3y1 + 4y2 y′ 2 = −y1 + y2 y′ 2 = −y1 + y2 diﬀerenciálegyenlet-rendszert y1(0) = 1, y2(0) = 0 kezdeti feltétel mellett.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 11. feladatsor: Laplace-transzformáció 1. Határozzuk meg az f : (0, ∞) → függvény Laplace-transzformáltját. R a) f (x) = cos[2] _x_ b) _f_ (x) = 0 ha x < a  _x−a_ ha a ≤ _x < b_ _,_ _b−a_ 1 ha x ≥ _b_ ahol 0 < a < b. 2. Határozzuk meg Laplace-transzformációval az y[′′′]+y = 1 differenciálegyenlet y(0) = y[′](0) = _y[′′](0) = 0 kezdeti feltételt kielégítő megoldását._ 3. Laplace-transzformáció alkalmazásával oldjuk meg az xy[′′] + 2y[′] + xy = 0 differenciálegyenletet. 4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze. Az egyiket a t0 = 0 pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük. Ha y1, y2 jelöli az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó differenciálegyenlet-rendszer _y1[′′]_ [=][ y][2] _[−]_ _[y][1]_ [+][ f] _y2[′′]_ [=][ y][1] _[−]_ _[y][2][.]_ Hogyan mozognak a t0 = 0 pillanat után? ## További gyakorló feladatok 5. Határozzuk meg az f : (0, ∞) → függvény Laplace-transzformáltját. R a) f (x) = x[−][1][/][2] b) f (x) = sgn sin(πx) 6. Laplace-transzformáció segítségével határozzuk meg az y[′′] _−_ 2y[′] + _y = x differenciálegyenlet_ _y(0) = 0 y[′](0) = −1 kezdeti feltételt kielégítő megoldását._ 7. Laplace-transzformáció segítségével oldjuk meg az y[′′] + 2y[′] + 2y = 0 differenciálegyenletet _y(0) = 0, y[′](0) = 1 kezdeti feltétel mellett._ 8. Laplace-transzformáció alkalmazásával oldjuk meg az xy[′′] + 2y[′] _−_ _xy = 0 differenciálegyen-_ letet. 9. Laplace-transzformáció alkalmazásával oldjuk meg az _y1[′]_ [=][ −][3][y][1] [+ 4][y][2] _y2[′]_ [=][ −][y][1] [+][ y][2] differenciálegyenlet-rendszert y1(0) = 1, y2(0) = 0 kezdeti feltétel mellett. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">11. feladatsor: Laplace-transzformáció</span></b></p> <p style="top:130.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> : (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény Laplace-transzformáltját.</span></p> <p style="top:144.8pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:161.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span></p> <p style="top:206.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:180.9pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:189.9pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:192.9pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:195.8pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:213.8pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:216.8pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:219.8pt;left:165.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:189.3pt;left:174.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:189.3pt;left:204.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x < a</span></i></p> <p style="top:205.2pt;left:175.5pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></p> <p style="top:214.0pt;left:176.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">b</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">a</span></i></p> <p style="top:206.7pt;left:204.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x < b</span></i></p> <p style="top:224.0pt;left:174.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:224.0pt;left:204.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≥</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">b</span></i></p> <p style="top:206.2pt;left:273.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:250.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> < a < b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:266.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Határozzuk meg Laplace-transzformációval az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></p> <p style="top:281.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását.</span></p> <p style="top:297.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Laplace-transzformáció alkalmazásával oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyen-</span></p> <p style="top:312.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">letet.</span></p> <p style="top:328.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze.</span></p> <p style="top:328.7pt;left:441.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az egyiket a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:343.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük.</span></p> <p style="top:343.1pt;left:464.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelö-</span></p> <p style="top:357.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">li az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó</span></p> <p style="top:372.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer</span></p> <p style="top:398.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup></p> <p style="top:404.6pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup></p> <p style="top:415.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup></p> <p style="top:422.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:442.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hogyan mozognak a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pillanat után?</span></p> <p style="top:475.0pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:499.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> : (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény Laplace-transzformáltját.</span></p> <p style="top:513.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:530.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sgn sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:546.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Laplace-transzformáció segítségével határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span></p> <p style="top:560.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását.</span></p> <p style="top:577.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Laplace-transzformáció segítségével oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span></p> <p style="top:591.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:608.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Laplace-transzformáció alkalmazásával oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyen-</span></p> <p style="top:622.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">letet.</span></p> <p style="top:639.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Laplace-transzformáció alkalmazásával oldjuk meg az</span></p> <p style="top:665.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:671.6pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:682.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:689.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:709.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> </div>	page_303.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 11. feladatsor: Laplace-transzformáció 1. Határozzuk meg az / : (0,o0) — R függvény Laplace-transzformáltját a) fír) — cor 1) 0 harzca fejzdes haaszeb 17 harzó ahol0 2 a 2. 2. Határozzuk meg Laplace-transzformációval az y"-y. w"(0) — 0 kezdeti felté 3. Laplace-transzformáció alkalmazásával oldjuk meg az 1" 4-24 4-2y — 0 differenciálegyer letet elt kielégítő megoldását. 4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze. Az egyiket a t9 — 0 pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük. Ha m.y2 jelő- W az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó dilferenciálegyet dszer ú [ DEE m. Hogyan mozognak a fg — 0 pillanat után? "További gyakorló feladatok 5. Határozzuk meg az / : (0,50) — R függvény Laplace-transzformáltját bEE 1) fér) — sznsinlrz) 10) — 0 w(0) — —1 kezdeti feltételt kielégítő megoldását el oldjuk meg az 4 424 4. 2y — 0 differenciálegyenletet el mellett. 244 y — x diferenciálegyenlet 7. Laplace-transzformáció segítsé. w(0) — 0. v(0) — 1 kezdeti felti 8. Laplace-transzformáció alkalmazásával oldjuk meg az 21/ 4- 2W — xy — 0 differenciálegyet letet 9. Laplace-transzformáció alkalmazásával oldjuk meg az W —3m 4492 B tn dilferenciálegyet kezdeti feltétel mellett v2(0)
Tehát y(x) = y0 sinh x sinh x Tehát y(x) = y0 x megoldás. Ez nem az általános megoldás, egy másikat kereshetünk például a Wronski-determináns segítségével. W ′ = −2 xW alapján W(x) = C x2. c(x)sinh x x 2 xW alapján W(x) = C x2. c(x)sinh x x akkor megoldás, ha sinh x !2 C x2 = c′(x) azaz c′(x) = C sinh2 x. Ebből c(x) = C tanh x + C2, és így egy másik lineárisan független megoldás cosh x x . Az egyenlet általános megoldása y(x) = Asinh x x + B cosh x x 9. Laplace-transzformáció alkalmazásával oldjuk meg az y′ 1 y′ 1 = −3y1 + 4y2 y′ 2 = −y1 + y2 y′ 2 = −y1 + y2 diﬀerenciálegyenlet-rendszert y1(0) = 1, y2(0) = 0 kezdeti feltétel mellett. Megoldás. Legyen Lyi = Yi, ekkor (Ly′ 1 y′ 1)(z) = zY1(z) −1 és (Ly′ 2 Megoldás. Legyen Lyi = Yi, ekkor (Ly′ 1)(z) = zY1(z) −1 és (Ly′ 2)(z) = zY2(z). Laplacetranszformáljuk az egyenletek mindkét oldalát: zY1(z) −1 = −3Y1(z) + 4Y2(z) zY2(z) = −Y1(z) + Y2(z), ez egy (algebrai) lineáris egyenletrendszer Y1(z) és Y2(z) ismeretlenekkel. A megoldás parciális törtekre bontva Y1(z) = −2 1 (z + 1)2 + 1 z + 1 Y2(z) = − 1 (z + 1)2, ezek inverz Laplace-transzformáltja y1(x) = −2xe−x + e−x y2(x) = −xe−x.	Tehát y(x) = y0 sinhx _x_ megoldás. Ez nem az általános megoldás, egy másikat kereshetünk például a Wronski-determináns segítségével. W _[′]_ = − [2] _C_ _x_ _[W][ alapján][ W]_ [(][x][) =] _x[2]_ [.][ c][(][x][)] [sinh]x _[ x]_ akkor megoldás, ha �2 _,_ _C_ _x[2][ =][ c][′][(][x][)]_ �sinh x _x_ azaz _C_ _c[′](x) =_ sinh[2] _x_ _[.]_ Ebből c(x) = C tanh x + C2, és így egy másik lineárisan független megoldás coshx _x_ . Az egyenlet általános megoldása _y(x) = A_ [sinh][ x] + B [cosh][ x] _._ _x_ _x_ 9. Laplace-transzformáció alkalmazásával oldjuk meg az _y1[′]_ [=][ −][3][y][1] [+ 4][y][2] _y2[′]_ [=][ −][y][1] [+][ y][2] differenciálegyenlet-rendszert y1(0) = 1, y2(0) = 0 kezdeti feltétel mellett. _Megoldás. Legyen Lyi = Yi, ekkor (Ly1[′]_ [)(][z][) =][ zY][1][(][z][)][ −] [1 és (][L][y]2[′] [)(][z][) =][ zY][2][(][z][). Laplace-] transzformáljuk az egyenletek mindkét oldalát: _zY1(z) −_ 1 = −3Y1(z) + 4Y2(z) _zY2(z) = −Y1(z) + Y2(z),_ ez egy (algebrai) lineáris egyenletrendszer Y1(z) és Y2(z) ismeretlenekkel. A megoldás parciális törtekre bontva 1 1 _Y1(z) = −2_ (z + 1)[2][ +] _z + 1_ 1 _Y2(z) = −_ (z + 1)[2] _[,]_ ezek inverz Laplace-transzformáltja _y1(x) = −2xe[−][x]_ + e[−][x] _y2(x) = −xe[−][x]._ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:57.6pt;left:162.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sinh</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:66.4pt;left:171.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:59.1pt;left:189.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldás. Ez nem az általános megoldás, egy másikat kereshetünk</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">például a Wronski-determináns segítségével.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:80.8pt;left:358.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">W</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alapján</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> W</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup></p> <p style="top:72.0pt;left:473.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">C</span></i></p> <p style="top:80.8pt;left:472.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sinh</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:80.8pt;left:524.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">akkor megoldás, ha</span></p> <p style="top:115.1pt;left:108.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></p> <p style="top:131.4pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:107.2pt;left:161.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:131.4pt;left:182.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:107.2pt;left:201.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:123.2pt;left:216.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:153.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span></p> <p style="top:182.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:174.3pt;left:159.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></p> <p style="top:191.2pt;left:147.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:214.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> tanh</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, és így egy másik lineárisan független megoldás</span></p> <p style="top:212.8pt;left:488.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">cosh</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></p> <p style="top:221.6pt;left:497.2pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:214.3pt;left:511.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az</span></p> <p style="top:228.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet általános megoldása</span></p> <p style="top:260.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:268.7pt;left:165.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:260.5pt;left:187.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cosh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:268.7pt;left:222.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:260.5pt;left:242.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:289.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Laplace-transzformáció alkalmazásával oldjuk meg az</span></p> <p style="top:316.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:322.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:333.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:339.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:360.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:379.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ekkor</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:385.7pt;left:276.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:385.7pt;left:410.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Laplace-</span></sup></p> <p style="top:394.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">transzformáljuk az egyenletek mindkét oldalát:</span></p> <p style="top:420.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:437.8pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:464.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ez egy (algebrai) lineáris egyenletrendszer</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ismeretlenekkel. A megoldás par-</span></p> <p style="top:478.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ciális törtekre bontva</span></p> <p style="top:507.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:499.1pt;left:182.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:515.4pt;left:165.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup></p> <p style="top:499.1pt;left:232.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:515.4pt;left:222.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></p> <p style="top:538.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:529.9pt;left:176.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:546.2pt;left:159.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:570.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ezek inverz Laplace-transzformáltja</span></p> <p style="top:596.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:614.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_304.png	példáni a Wronskédetermináns segítségével. WW" — —21W alapján IW(z) — £. elz)ánbz akkor megoldás, ha s.. az Ctanhz 4 Cy, és így egy másik lincárisan független megoldás egyenlet általános megoldása sinhz , peoshz v -— - Laplaco-transzformáció alkalmazásával oldjuk meg az 2 —3n 4492 n te dilferenciálegyenlet-rendszert y(0) — 1. y2(0) — 0 kezdeti feltétel mellett . Megoldás. Legyen £y, — X, ekkor (Cyi)(2) — 2Vi(2) — 1 és (£y)(e) — 2X2). Laplace EZő] 31 () 449429 NGK. ez egy (algebrai) lincáris egyes ciális törtekre bontva. etrendszer 1 (2) és Yí(2) ismeretlenekkel. A megoldás par- Eő] t] cezek inverz Laplace-transzforn mle) a 2ét e vle) 2e
3. Laplace-transzformáció alkalmazásával oldjuk meg az xy′′ + 2y′ + xy = 0 diﬀerenciálegyen- letet. Megoldás. Legyen Y = Ly és y(0) = y0, y′(0) = y′ 0 (paraméterek, mert az általános megoldást keressük). Ekkor L(xy)(z) = −Y ′(z) Ly′(z) = zY (z) −y0 L(xy′′)(z) = −d dz 0 d dz(z2Y (z) −zy0 −y′ y′ 0) = −2zY (z) −z2Y ′(z) + y0. Behelyettesítve 0 = −2zY (z) −z2Y ′(z) + y0 + 2(zY (z) −y0) −Y ′(z) = −(z2 + 1)Y ′(z) −y0, ami egy elsőrendű lineáris diﬀerenciálegyenlet Y (z)-re. Elég azonban −Y ′(z)-t kifejezni, mivel ez xy(x) Laplace-transzformáltja. −Y ′(z) = y0 1 + z2 alapján xy(x) = y0 sin x, tehát sin x y(x) = y0 x Ez nem az általános megoldás, mert csak egy paraméter van benne. A másik megoldást c(x)sin x x alakban keresve a 2c′(x) cos x + c′′(x) sin x = 0 egyenletet kapjuk, amiből c′′(x) c′′(x) c′(x) = −2cos x sin x c′(x) = 2 sin x (ln c′(x))′ = (−2 ln sin x)′ c′(x) = C sin2 x c(x) = −C cos x sin x , c′(x) = C sin2 x c(x) = −C cos x sin x tehát az általános megoldás y(x) = Asin x x + B cos x x 4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze. Az egyiket a t0 = 0 pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük. Ha y1, y2 jelöli az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó diﬀerenciálegyenlet-rendszer y′′ 1 y′′ 1 = y2 −y1 + f y′′ 2 = y1 −y2. y′′ 2 = y1 −y2. Hogyan mozognak a t0 = 0 pillanat után?	3. Laplace-transzformáció alkalmazásával oldjuk meg az xy[′′] + 2y[′] + xy = 0 differenciálegyenletet. _Megoldás. Legyen Y = Ly és y(0) = y0, y[′](0) = y0[′]_ [(paraméterek, mert az általános] megoldást keressük). Ekkor _L(xy)(z) = −Y_ _[′](z)_ _Ly[′](z) = zY (z) −_ _y0_ _L(xy[′′])(z) = −_ [d] 0[) =][ −][2][zY][ (][z][)][ −] _[z][2][Y][ ′][(][z][) +][ y][0][.]_ dz [(][z][2][Y][ (][z][)][ −] _[zy][0][ −]_ _[y][′]_ Behelyettesítve 0 = −2zY (z) − _z[2]Y_ _[′](z) + y0 + 2(zY (z) −_ _y0) −_ _Y_ _[′](z)_ = −(z[2] + 1)Y _[′](z) −_ _y0,_ ami egy elsőrendű lineáris differenciálegyenlet Y (z)-re. Elég azonban −Y _[′](z)-t kifejezni,_ mivel ez xy(x) Laplace-transzformáltja. _−Y_ _[′](z) =_ _y0_ 1 + z[2] alapján xy(x) = y0 sin x, tehát sin x _y(x) = y0_ _x_ Ez nem az általános megoldás, mert csak egy paraméter van benne. A másik megoldást _c(x)_ [sin][ x] alakban keresve a _x_ 2c[′](x) cos x + c[′′](x) sin x = 0 egyenletet kapjuk, amiből _c[′′](x)_ _c[′](x) [=][ −][2cos]sin x[ x]_ (ln c[′](x))[′] = (−2 ln sin x)[′] _C_ _c[′](x) =_ sin[2] _x_ _c(x) = −C_ [cos][ x] sin x [,] tehát az általános megoldás _y(x) = A_ [sin][ x] + B [cos][ x] _._ _x_ _x_ 4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze. Az egyiket a t0 = 0 pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük. Ha y1, y2 jelöli az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó differenciálegyenlet-rendszer _y1[′′]_ [=][ y][2] _[−]_ _[y][1]_ [+][ f] _y2[′′]_ [=][ y][1] _[−]_ _[y][2][.]_ Hogyan mozognak a t0 = 0 pillanat után? -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Laplace-transzformáció alkalmazásával oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyen-</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">letet.</span></p> <p style="top:92.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:98.8pt;left:358.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(paraméterek, mert az általános</span></sup></p> <p style="top:107.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldást keressük). Ekkor</span></p> <p style="top:132.7pt;left:111.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:150.1pt;left:124.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:174.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup></p> <p style="top:183.1pt;left:183.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zy</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:181.0pt;left:287.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:203.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Behelyettesítve</span></p> <p style="top:228.0pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:245.5pt;left:117.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:270.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ami egy elsőrendű lineáris diﬀerenciálegyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re. Elég azonban</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-t kifejezni,</span></p> <p style="top:285.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mivel ez</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Laplace-transzformáltja.</span></p> <p style="top:313.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:305.7pt;left:170.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:322.0pt;left:159.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:344.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alapján</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:374.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:366.7pt;left:155.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:383.0pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:403.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez nem az általános megoldás, mert csak egy paraméter van benne. A másik megoldást</span></p> <p style="top:417.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:425.1pt;left:105.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></p> <p style="top:417.8pt;left:120.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakban keresve a</span></p> <p style="top:443.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:469.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletet kapjuk, amiből</span></p> <p style="top:492.5pt;left:126.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:508.8pt;left:127.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:508.8pt;left:186.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></p> <p style="top:526.4pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 ln sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:550.9pt;left:130.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:542.9pt;left:180.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></p> <p style="top:559.1pt;left:170.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:576.4pt;left:132.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:584.6pt;left:189.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:605.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát az általános megoldás</span></p> <p style="top:635.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:643.3pt;left:162.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:635.1pt;left:181.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i></sup></p> <p style="top:643.3pt;left:212.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:635.1pt;left:228.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:663.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze.</span></p> <p style="top:663.6pt;left:441.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az egyiket a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:678.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük.</span></p> <p style="top:678.1pt;left:464.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelö-</span></p> <p style="top:692.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">li az egyes testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó</span></p> <p style="top:707.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer</span></p> <p style="top:732.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup></p> <p style="top:738.7pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup></p> <p style="top:750.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup></p> <p style="top:756.2pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Hogyan mozognak a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> t</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pillanat után?</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_306.png	3. Laplace-transzformáció alkalmazásával oldjuk meg az 1/ 4-2y4-zy — 0 differenciálegyen- tetet .Megoldás. Legyen Y — £y és y(0) — o. V(0) — 16 (paraméterek, mert az általános írh re EV -V £GYIAY -- LEVEA -9 V()— V) 4 n. Behelyettesítve Y (2) 4 0 4 2Y (—) Y ) ami egy elsőrendű lncáris diflerenciálegyenlet Y(2)-re. Elég azonban —Y(2)-t kifejezni, mivel ez zyíz) Laplace-transzformáltja. m Y alapján 2y(r) — yosin , tehát v Ez nem az általános megoldás, mert csak egy paraméter van benne. A másik megoldást elzjénz alakban keresve a 2íejoszá íránz-0 egyenletet kapjuk, amiből díz) , eosz e 7 7 ánz aY - (-2lnsinz) ele) a -0TS tehát az általános megoldás sinz , pe0sz EŐ] 4. Két nyugalomban lévő testet kezdetben nyújtatlan rugó köt össze. Az egyiket a t — 0 pillanatban a rugóval párhuzamos állandó erővel gyorsítani kezdjük. Ha m.w2 jelő- 1 az egyes differenciálegyenlet-rendszer testek kezdeti helytől számított elmozdulását, akkor a mozgást meghatározó DEE VY. Hogyan mozognak a £ — 0 pillanat után?
Megoldás. Legyen Y1 = Ly1 és Y2 = Ly2. A kezdeti feltétel y1(0) = y′ 1 2 2 y′ 1(0) = y2(0) = y′ 2 Megoldás. Legyen Y1 = Ly1 és Y2 = Ly2. A kezdeti feltétel y1(0) = y′ 1(0) = y2(0) = y′ 2(0) = 0, emiatt (Ly′′ 1)(z) = z2Y1(z) és (Ly′′ 2)(z) = z2Y2(z). Laplace-transzformáljuk az egyenletek 0, emiatt (Ly′′ 1)(z) = z2Y1(z) és (Ly′′ 2)(z) = z2Y2(z). Laplace-transzformáljuk az egyenletek mindkét oldalát: y′′ 1)(z) = z2Y1(z) és (Ly′′ 2 z2Y1(z) = Y2(z) −Y1(z) + f z z2Y2(z) = Y1(z) −Y2(z). Ez egy lineáris egyenletrendszer Y1(z)-re és Y2(z)-re, a megoldás parciális törtekre bontva 2 Y1(z) = f(1 + z2) 4 z3(z2 + 2) = f Y1(z) = f(1 + z2) z3(z2 + 2) 2 z3 + 1 z 1 z − z z2 + 2 z 2 4 Y2(z) = f z3(z2 + 2) = f 2 z3 −1 z 1 z + z z2 + 2 A kezdetiérték-probléma megoldását inverz Laplace-transzformációval kapjuk: y1(t) = f 4 y2(t) = f 4 t2 + 1 −cos( t2 −1 + cos( √ 2t) √ . 2t) További gyakorló feladatok 5. Határozzuk meg az f : (0, ∞) →R függvény Laplace-transzformáltját. a) f(x) = x−1/2 b) f(x) = sgn sin(πx) Megoldás. a) A Laplace-transzformált létezik, ha z > 0, mivel f integrálható (0, 1]-en és korlátos [1, ∞)-en. zx = u2, dx = 2u z du helyettesítéssel √π √z . Z ∞ Z ∞ e−u2 Z ∞ e−zx √x dx = e−zx 2u √z du = 2 √z e−u2 du = b) f korlátos, tehát létezik Laplace-transzformált a pozitív valós részű számok félsíkján, de \|f(x)\| majdnem mindenhol 1, tehát Re z ≤0-ra nem létezik. A függvény értéke 1 ha 2k < x < 2k + 1 és −1 ha 2k + 1 < x < 2k + 2 (k ∈Z), tehát Lf(z) = Z ∞ e−zxf(x) dx Z 2k+1 2k e−zx dx − Z 2k+1 ! Z 2k+2 2k+1 e−zx dx Z 2k+2 k=0 e−2kz −e−(2k+1)z −e−(2k+1)z −e−(2k+2)z z z z k=0 e−2kz 1 −2e−z + e−2z k=0 z = (1 −e−z)2 z 1 1 −e−2z . 6. Laplace-transzformáció segítségével határozzuk meg az y′′ −2y′ +y = x diﬀerenciálegyenlet y(0) = 0 y′(0) = −1 kezdeti feltételt kielégítő megoldását.	_Megoldás. Legyen Y1 = Ly1 és Y2 = Ly2. A kezdeti feltétel y1(0) = y1[′]_ [(0) =][ y][2][(0) =][ y]2[′] [(0) =] 0, emiatt (Ly1[′′][)(][z][) =][ z][2][Y][1][(][z][) és (][L][y]2[′′][)(][z][) =][ z][2][Y][2][(][z][). Laplace-transzformáljuk az egyenletek] mindkét oldalát: _z[2]Y1(z) = Y2(z) −_ _Y1(z) +_ _[f]_ _z_ _z[2]Y2(z) = Y1(z) −_ _Y2(z)._ Ez egy lineáris egyenletrendszer Y1(z)-re és Y2(z)-re, a megoldás parciális törtekre bontva _Y1(z) =_ _[f]_ [(1 +][ z][2][)] _z[3](z[2]_ + 2) [=][ f]4 _f_ _Y2(z) =_ _z[3](z[2]_ + 2) [=][ f]4 � 2 _z_ _z[3][ + 1]z_ _[−]_ _z[2]_ + 2 � 2 _z_ _z[3][ −]_ _z[1]_ [+] _z[2]_ + 2 � � _._ A kezdetiérték-probléma megoldását inverz Laplace-transzformációval kapjuk: 2t)� 2t)� _._ _y1(t) =_ _[f]_ 4 _y2(t) =_ _[f]_ 4 _√_ �t[2] + 1 − cos( _√_ �t[2] _−_ 1 + cos( ## További gyakorló feladatok 5. Határozzuk meg az f : (0, ∞) → függvény Laplace-transzformáltját. R a) f (x) = x[−][1][/][2] b) f (x) = sgn sin(πx) _Megoldás._ a) A Laplace-transzformált létezik, ha z > 0, mivel f integrálható (0, 1]-en és korlátos [1, ∞)-en. zx = u[2], dx = [2][u] _z_ [d][u][ helyettesítéssel] � _∞_ _e[−][zx]_ � _∞_ _e[−][u][2]_ 2u _√_ _x =_ _√_ _u = √[2]_ 0 _x d_ 0 _u_ _z d_ _z_ _√_ � _∞_ _π_ _e[−][u][2]_ du = _√_ 0 _z ._ b) f korlátos, tehát létezik Laplace-transzformált a pozitív valós részű számok félsíkján, de \|f (x)\| majdnem mindenhol 1, tehát Re z ≤ 0-ra nem létezik. A függvény értéke 1 ha 2k < x < 2k + 1 és −1 ha 2k + 1 < x < 2k + 2 (k ∈ ), tehát Z � _∞_ _Lf_ (z) = _e[−][zx]f_ (x) dx 0 _∞_ �� 2k+1 � 2k+2 � = � _e[−][zx]_ dx − _k=0_ 2k 2k+1 _[e][−][zx][ d][x]_ _∞_ �e−2kz − _e−(2k+1)z_ = � _−_ _[e][−][(2][k][+1)][z][ −]_ _[e][−][(2][k][+2)][z]_ _z_ _z_ _k=0_ _∞_ _e[−][2][kz]_ = � �1 − 2e[−][z] + e[−][2][z][�] _z_ _k=0_ � 1 = [(1][ −] _[e][−][z][)][2]_ _z_ 1 − _e[−][2][z][ .]_ 6. Laplace-transzformáció segítségével határozzuk meg az y[′′] _−_ 2y[′] + _y = x differenciálegyenlet_ _y(0) = 0 y[′](0) = −1 kezdeti feltételt kielégítő megoldását._ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A kezdeti feltétel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:65.2pt;left:424.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:65.2pt;left:506.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></sup></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, emiatt</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup></p> <p style="top:79.7pt;left:144.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup></p> <p style="top:79.7pt;left:257.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Laplace-transzformáljuk az egyenletek</span></sup></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mindkét oldalát:</span></p> <p style="top:117.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup></p> <p style="top:125.5pt;left:243.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:139.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:166.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez egy lineáris egyenletrendszer</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re, a megoldás parciális törtekre bontva</span></p> <p style="top:199.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:207.7pt;left:150.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup></p> <p style="top:207.7pt;left:219.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:186.6pt;left:229.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span></p> <p style="top:207.7pt;left:237.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup></p> <p style="top:207.7pt;left:265.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup></p> <p style="top:191.4pt;left:300.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:207.7pt;left:288.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></p> <p style="top:186.6pt;left:320.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:231.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:223.0pt;left:171.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></p> <p style="top:239.3pt;left:150.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2) </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i></sup></p> <p style="top:239.3pt;left:219.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:218.1pt;left:229.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span></p> <p style="top:239.3pt;left:237.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:239.3pt;left:265.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:223.0pt;left:300.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:239.3pt;left:288.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></p> <p style="top:218.1pt;left:320.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:231.1pt;left:329.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:263.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdetiérték-probléma megoldását inverz Laplace-transzformációval kapjuk:</span></p> <p style="top:295.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup></p> <p style="top:303.3pt;left:147.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:285.2pt;left:157.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:295.1pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span></p> <p style="top:284.8pt;left:227.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:295.1pt;left:237.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:285.2pt;left:252.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:323.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i></sup></p> <p style="top:331.9pt;left:147.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:313.7pt;left:157.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:323.7pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + cos(</span></p> <p style="top:313.4pt;left:227.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:323.7pt;left:237.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">t</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:313.7pt;left:252.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:323.7pt;left:259.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:357.5pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:381.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> : (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény Laplace-transzformáltját.</span></p> <p style="top:396.0pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">/</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:412.5pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sgn sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">πx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:431.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:446.4pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) A Laplace-transzformált létezik, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, mivel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> integrálható</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1]</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-en és korlátos</span></p> <p style="top:460.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">[1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-en.</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zx</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup></p> <p style="top:468.1pt;left:232.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítéssel</span></sup></p> <p style="top:484.5pt;left:126.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ∞</span></i></p> <p style="top:508.7pt;left:132.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:488.9pt;left:149.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">zx</span></i></sup></p> <p style="top:496.6pt;left:151.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:484.5pt;left:202.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ∞</span></i></p> <p style="top:508.7pt;left:208.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:488.9pt;left:224.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:505.2pt;left:232.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:488.9pt;left:250.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></p> <p style="top:496.6pt;left:248.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:496.6pt;left:298.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:484.5pt;left:317.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ∞</span></i></p> <p style="top:508.7pt;left:322.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:497.0pt;left:338.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">u</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:480.3pt;left:391.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i></p> <p style="top:496.6pt;left:392.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z .</span></i></p> <p style="top:530.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> korlátos, tehát létezik Laplace-transzformált a pozitív valós részű számok félsíkján,</span></p> <p style="top:544.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">de</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> majdnem mindenhol</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Re</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ra nem létezik.</span></p> <p style="top:558.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A függvény értéke</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k < x <</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> < x <</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">Z</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), tehát</span></p> <p style="top:589.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:576.7pt;left:173.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ∞</span></i></p> <p style="top:600.9pt;left:178.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:589.2pt;left:194.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">zx</span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:619.7pt;left:160.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:609.9pt;left:176.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:611.3pt;left:173.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:635.0pt;left:173.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:603.7pt;left:190.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span></p> <p style="top:631.4pt;left:203.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></p> <p style="top:619.7pt;left:230.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">zx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:607.2pt;left:281.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2</span></p> <p style="top:631.4pt;left:287.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">zx</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup></p> <p style="top:603.7pt;left:350.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:655.2pt;left:160.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:645.5pt;left:176.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:646.9pt;left:173.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:670.6pt;left:173.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:639.3pt;left:190.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"> </span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kz</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1)</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></p> <p style="top:663.4pt;left:238.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:655.2pt;left:286.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+1)</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+2)</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup></p> <p style="top:663.4pt;left:347.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:639.3pt;left:401.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">!</span></p> <p style="top:690.3pt;left:160.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:680.6pt;left:176.5pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">∞</span></i></p> <p style="top:682.0pt;left:173.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:705.7pt;left:173.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">k</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=0</span></p> <p style="top:682.3pt;left:191.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">kz</span></i></sup></p> <p style="top:698.6pt;left:201.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:680.4pt;left:220.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:690.3pt;left:226.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></sup></p> <p style="top:725.1pt;left:160.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:733.3pt;left:196.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i></p> <p style="top:717.0pt;left:245.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:733.3pt;left:227.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> .</span></i></sup></p> <p style="top:756.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Laplace-transzformáció segítségével határozzuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet</span></p> <p style="top:770.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltételt kielégítő megoldását.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_307.png	. Megoldás. Les 0. emiatt (£y7)! mindkét oldalát: Yi — Cy és X4 — Lya. A kezdeti feltétel yi (0) — 1 (0) 2PV.(2) és (£yf)e) — V(). Laplace-transzformi G-10-191£ XA9-1G9-X0. jar fgf2,1 2 ARE 3) ette m L(E 41-—esívőn) Í íe ml 2 £(£-1 4 cosívő) "További gyakorló feladatok 5. Határozzuk meg az / : (0,00) — R függvény Laplace-transzformáltját. . Megoldás. a) A Laplace-transzformi L, öc)-en. 2r — 12. . létezik, ha 2 —. 0, mivel / integrálható (0,1j-en és korlátos 22 du helyettesítéssel ar ] 3) / korlátos, tehát létezik Laplace-transzformált a pozitív valós részű számok félsíkján, A függvény értéke 1 ha 2k c 7 22k 41 és -1 ha2k412r22k42(k€Z7),tehát duz erő2 főesnöga 6. Laplace-transzformáció segítségével határozzuk meg az 4" —2y/--y — x differenciálegyenlet w(0) — 0 w(0) — —1 kezdeti feltételt kielégítő megoldását 3
Megoldás. Legyen Y = Ly, ekkor Ly′(z) = zY (z) −y(0) = zY (z), Ly′′(z) = z2Y (z) − zy(0) −y′(0) = z2Y (z) + 1. Az egyenlet mindkét oldalát Laplace-transzformáljuk: z2Y (z) + 1 −2zY (z) + Y (z) = 1 z2, amiből Y (z) = 1 z2 −1 z2 −2z + 1 = −2 1 z −1 + 1 z2 1 z2 + 21 z 1 z. Ez az y(x) = −2ex + x + 2 Laplace-transzformáltja. 7. Laplace-transzformáció segítségével oldjuk meg az y′′ + 2y′ + 2y = 0 diﬀerenciálegyenletet y(0) = 0, y′(0) = 1 kezdeti feltétel mellett. Megoldás. Legyen Y (z) = Ly(z), ekkor Ly′(z) = zY (z) −y(0) = zY (z) és Ly′′(z) = z2Y (z) −zy(0) −y′(0) = z2Y (z) −1, tehát a megoldandó egyenlet z2Y (z) −1 + 2zY (z) + 2Y (z) = 0, amiből Y (z) = 1 z2 + 2z + 2 = 1 (z + 1)2 + 1. Ez az y(x) = e−x sin x függvény Laplace-transzformáltja. 8. Laplace-transzformáció alkalmazásával oldjuk meg az xy′′ + 2y′ −xy = 0 diﬀerenciálegyen- letet. Megoldás. Legyen Ly = Y és y(0) = y0, y′(0) = y′ 0, ekkor (Ly′)(z) = zY (z) −y0 (Ly′′)(z) = z2Y (z) −zy0 −y′ 0 L(xy)(z) = −Y ′(z) L(xy′′)(z) = −2zY (z) −z2Y ′(z) + y0 Az egyenlet mindkét oldalát Laplace-transzformáljuk: −2zY (z) −z2Y ′(z) + y0 + 2(zY (z) −y0) −(−Y ′(z)) = 0, amiből 2 −Y ′(z) = y0 z2 −1 = y0 2 1 z −1 −y0 1 z + 1, ennek inverz Laplace-transzformáltja y0 y0 2 2 ex −y0 y0 2 e−x = y0 sinh x.	_Megoldás. Legyen Y = Ly, ekkor Ly[′](z) = zY (z) −_ _y(0) = zY (z), Ly[′′](z) = z[2]Y (z) −_ _zy(0) −_ _y[′](0) = z[2]Y (z) + 1. Az egyenlet mindkét oldalát Laplace-transzformáljuk:_ _z[2]Y (z) + 1 −_ 2zY (z) + Y (z) = [1] _z[2]_ _[,]_ amiből _Y (z) =_ _z1[2][ −]_ [1] 1 _z[2]_ _−_ 2z + 1 [=][ −][2]z − 1 [+ 1]z[2][ + 21]z _[.]_ Ez az y(x) = −2e[x] + x + 2 Laplace-transzformáltja. 7. Laplace-transzformáció segítségével oldjuk meg az y[′′] + 2y[′] + 2y = 0 differenciálegyenletet _y(0) = 0, y[′](0) = 1 kezdeti feltétel mellett._ _Megoldás. Legyen Y (z) = Ly(z), ekkor_ _Ly[′](z) = zY (z) −_ _y(0) = zY (z)_ és _Ly[′′](z) = z[2]Y (z) −_ _zy(0) −_ _y[′](0) = z[2]Y (z) −_ 1, tehát a megoldandó egyenlet _z[2]Y (z) −_ 1 + 2zY (z) + 2Y (z) = 0, amiből 1 1 _Y (z) =_ _z[2]_ + 2z + 2 [=] (z + 1)[2] + 1[.] Ez az y(x) = e[−][x] sin x függvény Laplace-transzformáltja. 8. Laplace-transzformáció alkalmazásával oldjuk meg az xy[′′] + 2y[′] _−_ _xy = 0 differenciálegyen-_ letet. _Megoldás. Legyen Ly = Y és y(0) = y0, y[′](0) = y0[′]_ [, ekkor] (Ly[′])(z) = zY (z) − _y0_ (Ly[′′])(z) = z[2]Y (z) − _zy0 −_ _y0[′]_ _L(xy)(z) = −Y_ _[′](z)_ _L(xy[′′])(z) = −2zY (z) −_ _z[2]Y_ _[′](z) + y0_ Az egyenlet mindkét oldalát Laplace-transzformáljuk: _−2zY (z) −_ _z[2]Y_ _[′](z) + y0 + 2(zY (z) −_ _y0) −_ (−Y _[′](z)) = 0,_ amiből _y0_ 1 1 _−Y_ _[′](z) =_ _z[2]_ _−_ 1 [=][ y]2[0] _z −_ 1 _[−]_ _[y]2[0]_ _z + 1[,]_ ennek inverz Laplace-transzformáltja _y0_ 2 _[e][x][ −]_ _[y]2[0]_ _[e][−][x][ =][ y][0][ sinh][ x.]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ekkor</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az egyenlet mindkét oldalát Laplace-transzformáljuk:</span></p> <p style="top:105.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:113.2pt;left:268.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:134.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span></p> <p style="top:165.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:155.5pt;left:164.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:164.3pt;left:162.1pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:173.7pt;left:147.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:157.4pt;left:248.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:173.7pt;left:238.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span></sup></p> <p style="top:173.7pt;left:281.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 21</span></sup></p> <p style="top:173.7pt;left:315.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:196.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Laplace-transzformáltja.</span></p> <p style="top:216.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Laplace-transzformáció segítségével oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span></p> <p style="top:230.4pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:249.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ekkor</span></p> <p style="top:276.1pt;left:108.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:305.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></p> <p style="top:334.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:361.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a megoldandó egyenlet</span></p> <p style="top:387.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:413.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span></p> <p style="top:442.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:434.1pt;left:173.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:450.4pt;left:147.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:434.1pt;left:250.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:450.4pt;left:223.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:475.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ez az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> függvény Laplace-transzformáltja.</span></p> <p style="top:494.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Laplace-transzformáció alkalmazásával oldjuk meg az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyen-</span></p> <p style="top:509.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">letet.</span></p> <p style="top:528.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:534.8pt;left:328.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ekkor</span></sup></p> <p style="top:554.9pt;left:115.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:572.4pt;left:113.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zy</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:578.5pt;left:259.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:589.8pt;left:111.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:607.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">L</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:633.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az egyenlet mindkét oldalát Laplace-transzformáljuk:</span></p> <p style="top:659.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">zY</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) = 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:686.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amiből</span></p> <p style="top:714.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Y</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:706.6pt;left:169.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:722.9pt;left:159.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup></p> <p style="top:722.9pt;left:211.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:706.6pt;left:232.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:722.9pt;left:222.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup></p> <p style="top:722.9pt;left:267.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:706.6pt;left:288.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:722.9pt;left:278.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">z</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:745.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek inverz Laplace-transzformáltja</span></p> <p style="top:765.3pt;left:107.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></p> <p style="top:781.6pt;left:109.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup></p> <p style="top:781.6pt;left:148.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sinh</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x.</span></i></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_308.png	"Megoldás. Legyen Y zv(0) — y00) — eV( y. ekkor £y/(-) FYE 2Y 4 Y Y 2t Tz az víz) 2 —2E 42 42 Laplace-transzformáltja. Laplace-transzformáció se w(0) — 0. v " Megoldás. Legyen Y(2) — — 1 kezdeti feltétel mellett. víz), ekkor FIŐEGTDET ÉBST £y(e) 2 V()— 2v(0) — (0) — Y -1. tehát a megoldandó egyenlet FYE-14 2794 2(— 0. 1 1 MEE TEE NK EZVET! tetet "Megoldás. Legyen £y — (E éés y(0) — 19. 4(0) — 6. ekkor V9-m FYE - 2n—a6 Y 2YY 4 Az egyenlet mindkét oldalát Laplace-transzformáljuk: 2Y (2) — ÉY() 40 4 2GY(2) — y0) - (-Y )— 0. amiből v ennek ín erz Laplace-transzformáltja vosính . ) — y(0) — 2V). £Y) oldalát Laplace-transzformi tségével oldjuk meg az 4" 4-2Y 4 2y — 0 difforenciálegyenletet y — 0 differenciálegyen-
8. Oldjuk meg az y′ = y −1 3 diﬀerenciálegyenlet-rendszert y(0) = (3, 2) kezdeti feltétel mellett. 9. Oldjuk meg az     10 −19 13 1 −1 1 −9 18 −12 y′ =  y 2 diﬀerenciálegyenlet-rendszert y(0) = (1, 0, 1) kezdeti feltétel mellett. 10. Oldjuk meg az     5 −3 4 6 −3 3 −1 1 −2 y′ =  y diﬀerenciálegyenlet-rendszert y(0) = (1, 1, −1) kezdeti feltétel mellett. 11. ∗Határozzuk meg az y′′ +Ω2y = 0 diﬀerenciálegyenlet-rendszer általános megoldását, ha Ω n × n méretű szimmetrikus pozitív deﬁnit mátrix. Mutassuk meg, hogy minden megoldás korlátos. 12. ∗Legyenek M, C, K n×n-es mátrixok, M invertálható, és tekintsük az My′′+Cy′+Ky = 0 másodrendű diﬀerenciálegyenlet-rendszert. a) Írjuk át elsőrendű egyenletrendszerré az y, y′ komponenseit tartalmazó (2n elemű) vek- torértékű függvényre nézve. b) Mutassuk meg, hogy az így kapott elsőrendű állandó együtthatós diﬀerenciálegyenlet- rendszer mátrixának sajátértékei a det(λ2M + λC + K) = 0 egyenlet gyökei. (Az ilyen típusú egyenletek neve kvadratikus sajátérték-probléma.) c) Tegyük fel, hogy M, C és K mindegyike pozitív deﬁnit. Mutassuk meg, hogy ekkor minden sajátérték valós része negatív. (Ebből következik, hogy minden megoldás 0-hoz tart amint x →∞.)	8. Oldjuk meg az y[′] = � 1 1� y _−1_ 3 differenciálegyenlet-rendszert y(0) = (3, 2) kezdeti feltétel mellett. 9. Oldjuk meg az y[′] =  10 _−19_ 13   1 _−1_ 1  y _−9_ 18 _−12_ differenciálegyenlet-rendszert y(0) = (1, 0, 1) kezdeti feltétel mellett. 10. Oldjuk meg az y[′] =  5 _−3_ 4   6 _−3_ 3  y _−1_ 1 _−2_ differenciálegyenlet-rendszert y(0) = (1, 1, −1) kezdeti feltétel mellett. 11. ∗ Határozzuk meg az y[′′] +Ω[2]y = 0 differenciálegyenlet-rendszer általános megoldását, ha Ω _n × n méretű szimmetrikus pozitív definit mátrix. Mutassuk meg, hogy minden megoldás_ korlátos. 12. ∗ Legyenek M, C, K n×n-es mátrixok, M invertálható, és tekintsük az M y[′′]+Cy[′]+Ky = 0 másodrendű differenciálegyenlet-rendszert. a) Írjuk át elsőrendű egyenletrendszerré az y, y[′] komponenseit tartalmazó (2n elemű) vektorértékű függvényre nézve. b) Mutassuk meg, hogy az így kapott elsőrendű állandó együtthatós differenciálegyenletrendszer mátrixának sajátértékei a det(λ[2]M + λC + K) = 0 egyenlet gyökei. (Az ilyen típusú egyenletek neve kvadratikus sajátérték-probléma.) c) Tegyük fel, hogy M, C és K mindegyike pozitív definit. Mutassuk meg, hogy ekkor minden sajátérték valós része negatív. (Ebből következik, hogy minden megoldás 0-hoz tart amint x →∞.) -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Oldjuk meg az</span></p> <p style="top:92.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:76.4pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:85.0pt;left:142.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:85.0pt;left:163.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:99.4pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:99.4pt;left:163.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:76.4pt;left:168.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:92.3pt;left:176.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:126.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:142.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Oldjuk meg az</span></p> <p style="top:182.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:160.6pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:178.2pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:168.0pt;left:140.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span></p> <p style="top:168.0pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">19</span></p> <p style="top:168.0pt;left:199.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13</span></p> <p style="top:182.4pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:182.4pt;left:166.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:182.4pt;left:202.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:196.9pt;left:138.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:196.9pt;left:168.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">18</span></p> <p style="top:196.9pt;left:194.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span></p> <p style="top:160.6pt;left:215.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:178.2pt;left:215.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:223.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:239.6pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Oldjuk meg az</span></p> <p style="top:279.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:257.6pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:275.1pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:265.0pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:265.0pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:265.0pt;left:193.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:279.4pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:279.4pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:279.4pt;left:193.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:293.9pt;left:138.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:293.9pt;left:168.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:293.9pt;left:189.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:257.6pt;left:204.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:275.1pt;left:204.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:320.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:336.5pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11.</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∗</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet-rendszer általános megoldását, ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span></p> <p style="top:351.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> méretű szimmetrikus pozitív deﬁnit mátrix. Mutassuk meg, hogy minden megoldás</span></p> <p style="top:365.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">korlátos.</span></p> <p style="top:381.9pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12.</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∗</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M, C, K n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-es mátrixok,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> invertálható, és tekintsük az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:396.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">másodrendű diﬀerenciálegyenlet-rendszert.</span></p> <p style="top:410.8pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) Írjuk át elsőrendű egyenletrendszerré az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">komponenseit tartalmazó (</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> elemű) vek-</span></p> <p style="top:425.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">torértékű függvényre nézve.</span></p> <p style="top:441.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) Mutassuk meg, hogy az így kapott elsőrendű állandó együtthatós diﬀerenciálegyenlet-</span></p> <p style="top:456.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rendszer mátrixának sajátértékei a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenlet gyökei. (Az ilyen</span></p> <p style="top:470.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">típusú egyenletek neve kvadratikus sajátérték-probléma.)</span></p> <p style="top:487.0pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) Tegyük fel, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M, C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mindegyike pozitív deﬁnit. Mutassuk meg, hogy ekkor</span></p> <p style="top:501.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">minden sajátérték valós része negatív. (Ebből következik, hogy minden megoldás</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-hoz</span></p> <p style="top:515.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tart amint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.)</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_309.png	10. 1. Oldjuk meg az l 1 1 3 differenciálegyer Oldjuk meg az po 9 8] Lx 1 lly t] dszert y(0) — (3,2) kezdeti feltétel mellett differenciálegyer Oldjuk meg az dszert y(0) — (1.0.1) kezdet feltétel mellett. differenciálegyer dszert y(0) — (1.1,—1) kezdeti feltétel mellett 1 Határozzuk meg az y" 4-?y — 0 díflerenciálegyenlet-rendszer általános megoldását, ha £ m x n méretű szimmetrikus pozitív definít mátrix. Mutassuk meg, hogy minden megoldás korlátos. " Legyenek M.C, K ncn-es mátrixok, M invertálható, és tekintsük az My"4-Cy"4-Ky. erenciálegyenlet a det32M 4-3C - K) — 0 egyenlet győkei. (Az ilye Tegyük fel, hogy ALC é rendszer mátrisának sajáti K mindegyike pozitív definit. Mutassuk meg, hogy ekkoör valós része negatív. (Ebből következi tart amint 5 — 00.) hogy minden megoldás 0-hoz
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 12. feladatsor: Lineáris állandó együtthatós egyenletrendszerek 1. Mi a Jordan-felbontása az A = −9 −7 mátrixnak? 2. Adjuk meg az A =    4 1 −1 −2 1 1    mátrix Jordan-felbontását. 3. Határozzuk meg az y′ 1 y′ 1 = 5y1 + 4y2 y′ 2 = −9y1 −7y y′ 2 = −9y1 −7y2 diﬀerenciálegyenlet-rendszer általános megoldását. 4. Oldjuk meg az y′ 1 y′ 1 = −y1 + 2y2 y′ 2 = −2y1 −y2 y′ 2 = −2y1 −y2 diﬀerenciálegyenlet-rendszert y1(0) = 1, y2(0) = −1 kezdeti feltétel mellett. 5. Oldjuk meg az y′ 1 y′ 1 = −2y1 + y2 y′ 2 = −2y2 + y3 y′ 2 = −2y2 + y3 y′ 3 = −2y3 y′ 3 = −2y3 diﬀerenciálegyenlet-rendszert y1(0) = 0, y2(0) = 0, y3(0) = 1 kezdeti feltétel mellett. További gyakorló feladatok 6. Határozzuk meg az      4 2 −3 2 0 1 0 0 5 2 −4 2 −1 −2 1 −1  A = mátrix Jordan-féle normálalakját. 7. Határozzuk meg az y′ 1 y′ 1 = 2y1 + 3y2 y′ 2 = −3y1 + 2y y′ 2 = −3y1 + 2y2 diﬀerenciálegyenlet-rendszer általános megoldását.	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 12. feladatsor: Lineáris állandó együtthatós egyenletrendszerek 1. Mi a Jordan-felbontása az _A =_ � 5 4 � _−9_ _−7_ mátrixnak? 2. Adjuk meg az _A =_  4 1 _−1_ −2 1 1  2 1 1 mátrix Jordan-felbontását. 3. Határozzuk meg az _y1[′]_ [= 5][y][1] [+ 4][y][2] _y2[′]_ [=][ −][9][y][1] _[−]_ [7][y][2] differenciálegyenlet-rendszer általános megoldását. 4. Oldjuk meg az _y1[′]_ [=][ −][y][1] [+ 2][y][2] _y2[′]_ [=][ −][2][y][1] _[−]_ _[y][2]_ differenciálegyenlet-rendszert y1(0) = 1, y2(0) = −1 kezdeti feltétel mellett. 5. Oldjuk meg az _y1[′]_ [=][ −][2][y][1] [+][ y][2] _y2[′]_ [=][ −][2][y][2] [+][ y][3] _y3[′]_ [=][ −][2][y][3] differenciálegyenlet-rendszert y1(0) = 0, y2(0) = 0, y3(0) = 1 kezdeti feltétel mellett. ## További gyakorló feladatok 6. Határozzuk meg az _A =_  4 2 _−3_ 2  0 1 0 0 5 2 _−4_ 2   _−1_ _−2_ 1 _−1_ mátrix Jordan-féle normálalakját. 7. Határozzuk meg az _y1[′]_ [= 2][y][1] [+ 3][y][2] _y2[′]_ [=][ −][3][y][1] [+ 2][y][2] differenciálegyenlet-rendszer általános megoldását. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">12. feladatsor: Lineáris állandó együtthatós</span></b></p> <p style="top:106.3pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">egyenletrendszerek</span></b></p> <p style="top:142.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Mi a Jordan-felbontása az</span></p> <p style="top:170.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:154.3pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:162.9pt;left:141.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:162.9pt;left:166.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:177.4pt;left:136.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:177.4pt;left:161.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> <p style="top:154.3pt;left:177.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:200.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrixnak?</span></p> <p style="top:216.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Adjuk meg az</span></p> <p style="top:253.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:231.9pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:249.4pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:239.3pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:239.3pt;left:162.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:239.3pt;left:178.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:253.7pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:253.7pt;left:162.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:253.7pt;left:183.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:268.2pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:268.2pt;left:162.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:268.2pt;left:183.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:231.9pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:249.4pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:290.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrix Jordan-felbontását.</span></p> <p style="top:307.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Határozzuk meg az</span></p> <p style="top:330.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:336.7pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 5</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:348.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:354.1pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:371.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását.</span></p> <p style="top:387.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. Oldjuk meg az</span></p> <p style="top:411.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:417.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:428.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:434.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:452.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:468.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Oldjuk meg az</span></p> <p style="top:492.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:498.1pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:509.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:515.5pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:526.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:533.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:550.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:582.6pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:606.6pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozzuk meg az</span></p> <p style="top:650.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:622.9pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:640.5pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:629.0pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:629.0pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:629.0pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:629.0pt;left:217.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:643.5pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:643.5pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:643.5pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:643.5pt;left:217.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:657.9pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:657.9pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:657.9pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:657.9pt;left:217.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:672.4pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:672.4pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:672.4pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:672.4pt;left:212.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:622.9pt;left:228.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:640.5pt;left:228.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:694.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrix Jordan-féle normálalakját.</span></p> <p style="top:711.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Határozzuk meg az</span></p> <p style="top:734.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:740.9pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:752.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:758.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer általános megoldását.</span></p> </div>	page_310.png	Matematika A3 gyakorlat 12. feladatsor: Lineáris állandó együtthatós egyenletrendszerek 1. Mi a Jordan-felbontása az mátrixnak? 2. Adjuk meg az mátrix Jordan-felbontását. 3. Határozzuk meg az W őmn t 492 2— —9 — Ty2 dilferenciák egyenlet-rendszer általános megoldását 4. Oldjuk meg az W 2m W. 2m — dilferenciálegyenk 1 kezdeti feltétel mellett. endszert y(0) — 1. y2(0) 5. Oldjuk meg az ME 2n tn 2t9 2 FA úz dilferenciálegyenlet endszert y1(0) — 0. ya(0) — 0. ys(0) — 1 kezdeti feltétel mellett "További gyakorló feladatok 6. Határozzuk meg az M2m 43 W9——3m 42 dilferenciálegyenk endszer általános megoldását.
c) Vezessük be a következő jelöléseket, ahol v ∈Cn: m(v) = ⟨v, Mv⟩ c(v) = ⟨v, Cv⟩ k(v) = ⟨v, Kv⟩. (A skalárszorzat komplex értelemben értendő és a második változóban lineáris, tehát v = (v1, . . . , vn) és w = (w1, . . . , wn) esetén ⟨v, w⟩= v1w1 + · · · + vnwn.) Ekkor m, c, k nemnegatív függvények, és mindegyik csak a nullvektoron veszi fel a 0 értéket. Ha λ ∈C olyan, hogy det(K + λC + λ2M) = 0, akkor létezik olyan v ̸= 0, amire (K + λC + λ2M)v = 0, következésképp 0 = ⟨v, (K + λC + λ2M)v⟩= k(v) + λc(v) + λ2m(v), tehát λ egy olyan másodfokú egyenlet gyöke, aminek minden együtthatója pozitív. A megoldóképlet alapján q c(v)2 −4m(v)k(v) 2m(v) λ = −c(v) + vagy q c(v)2 −4m(v)k(v) 2m(v) . λ = −c(v) − Mindkét szám valós része negatív, mert ha c(v)2 < 4m(v)k(v), akkor Re λ = −c(v) 2m(v c(v) 2m(v), ha viszont c(v)2 ≥4m(v)k(v), akkor q c(v)2 −4m(v)k(v) < c(v). q c(v)2 −4m(v)k(v)	c) Vezessük be a következő jelöléseket, ahol v ∈ : C[n] _m(v) = ⟨v, M_ v⟩ _c(v) = ⟨v, Cv⟩_ _k(v) = ⟨v, Kv⟩._ (A skalárszorzat komplex értelemben értendő és a második változóban lineáris, tehát v = (v1, . . ., vn) és w = (w1, . . ., wn) esetén ⟨v, w⟩ = v1w1 + · · · + vnwn.) Ekkor m, c, k nemnegatív függvények, és mindegyik csak a nullvektoron veszi fel a 0 értéket. Ha λ ∈ olyan, hogy det(K + λC + λ[2]M ) = 0, akkor létezik olyan v ̸= 0, amire C (K + λC + λ[2]M )v = 0, következésképp 0 = ⟨v, (K + λC + λ[2]M )v⟩ = k(v) + λc(v) + λ[2]m(v), tehát λ egy olyan másodfokú egyenlet gyöke, aminek minden együtthatója pozitív. A megoldóképlet alapján � _−c(v) +_ _λ =_ _c(v)[2]_ _−_ 4m(v)k(v) 2m(v) vagy � _−c(v) −_ _c(v)[2]_ _−_ 4m(v)k(v) _λ =_ _._ 2m(v) Mindkét szám valós része negatív, mert ha c(v)[2] _< 4m(v)k(v), akkor Re λ = −_ _[c][(][v][)]_ 2m(v) [,] � ha viszont c(v)[2] _≥_ 4m(v)k(v), akkor _c(v)[2]_ _−_ 4m(v)k(v) _< c(v)._ �� -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) Vezessük be a következő jelöléseket, ahol</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">C</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">:</span></p> <p style="top:85.5pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ⟨</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, M</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">⟩</span></i></p> <p style="top:102.9pt;left:132.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ⟨</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, C</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">⟩</span></i></p> <p style="top:120.4pt;left:130.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ⟨</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, K</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">⟩</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:146.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(A skalárszorzat komplex értelemben értendő és a második változóban lineáris, tehát</span></p> <p style="top:161.2pt;left:97.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">v</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ⟨</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> w</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">⟩</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> · · ·</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> v</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">w</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.) Ekkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> m, c, k</span></i></p> <p style="top:175.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">nemnegatív függvények, és mindegyik csak a nullvektoron veszi fel a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értéket.</span></p> <p style="top:190.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">C</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> olyan, hogy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor létezik olyan</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amire</span></p> <p style="top:204.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, következésképp</span></p> <p style="top:230.9pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ⟨</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">⟩</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λc</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:257.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egy olyan másodfokú egyenlet gyöke, aminek minden együtthatója pozitív. A</span></p> <p style="top:271.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldóképlet alapján</span></p> <p style="top:309.8pt;left:130.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:299.9pt;left:154.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) +</span></p> <p style="top:289.5pt;left:199.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:299.9pt;left:209.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:318.0pt;left:213.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:342.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vagy</span></p> <p style="top:383.0pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:373.1pt;left:150.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:362.7pt;left:195.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:373.1pt;left:205.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:391.2pt;left:209.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:383.0pt;left:302.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:418.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mindkét szám valós része negatív, mert ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"><</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Re</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">c</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">v</span></b></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup></p> <p style="top:426.0pt;left:510.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">m</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">v</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:438.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ha viszont</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">≥</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor</span></p> <p style="top:428.3pt;left:290.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:428.2pt;left:293.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:438.6pt;left:303.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">m</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:428.3pt;left:399.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> < c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> </div>	page_311.png	e) Vezcssük be a következő jelöléseket, ahol v € C": m(v) — (v.Mv) elv) — (v.Cv) kív) — (v.Kv). W (..s.,ts) és W3 (.. ,194) esetén (v, w) — n -- 4-Ttey.) Ekkor m.e, k Ha 3 € C olyan, hogy det(K 4. XC 4 X2M) — 0. akkor létezik olyan v 24 0, amire (K X3C 4 XÉMJv — 0. következésképp 0 (v.(K 4 XC 4XE MJV) — kív) 4 Xelv) 4 Vmlv), tehát X egy olyan másodfokú egyenlet győke, aminel megoldóképlet alapján minden együtthatója pozitív. A elv) 4 eVE mlvJEv] EZTŐ] vagy -elv) — él7 — ámlv)klv) 2m(v) E AMindkét szám valós része negatív, mert ha elv)? £ dm(v)k(v). akkor Red — ha viszont elv)? 2 dmv)kív). akkor eE — Im9EG] 2 elv)
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 12. feladatsor: Lineáris állandó együtthatós egyenletrendszerek (megoldás) 1. Mi a Jordan-felbontása az A = −9 −7 mátrixnak? Megoldás. det(A −λI) = λ2 + 2λ + 1 gyöke λ = −1 (kétszeres). # " x −9 −6 " # x1 x2 összes megoldása x1 = 2, x2 = −3 többszöröse, ezek a sajátvektorok. # " x −9 −6 " # x1 x2 −3 egy megoldása x1 = 1, x2 = −1, tehát A = S " −1 1 0 −1 S−1, ahol S = S−1 = " # −1 −1 −3 −1 2. Adjuk meg az A =    4 1 −1 −2 1 1    mátrix Jordan-felbontását. Megoldás. det(A −λI) = −λ3 + 6λ2 −12λ + 8 = 0 megoldása λ = 2 (háromszoros gyök)      2 1 −1 −2 −1 1 2 1 −1 · x x     x1 x2 x3 =   0 0 megoldásai (x1, x2, x3) = α(1, 0, 2) + β(0, 1, 1) alakúak, tehát csak két lineárisan független sajátvektor van. A mátrix mindent (1, −1, 1) többszöröseibe visz, válasszuk ezt az egyik bázisvektornak, egy őse pl. (0, 1, 0), a harmadik lehet (0, 1, 1). Ebből A = S 0 0 S−1, 0 2 ahol S = −1 1 −1  0  S−1 =   2 1 −1 −1 0 1 .	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 12. feladatsor: Lineáris állandó együtthatós egyenletrendszerek (megoldás) 1. Mi a Jordan-felbontása az _A =_ � 5 4 � _−9_ _−7_ mátrixnak? _Megoldás. det(A −_ _λI) = λ[2]_ + 2λ + 1 gyöke λ = −1 (kétszeres). � 6 4 ��x1� = �0� _−9_ _−6_ _x2_ 0 összes megoldása x1 = 2, x2 = −3 többszöröse, ezek a sajátvektorok. � 6 4 ��x1� = � 2 � _−9_ _−6_ _x2_ _−3_ egy megoldása x1 = 1, x2 = −1, tehát _A = S_ �−1 1 � _S[−][1],_ 0 _−1_ ahol �−1 _−1�_ _._ 3 2 _S =_ � 2 1 � _S[−][1]_ = _−3_ _−1_ 2. Adjuk meg az _A =_  4 1 _−1_ −2 1 1  2 1 1 mátrix Jordan-felbontását. _Megoldás. det(A −_ _λI) = −λ[3]_ + 6λ[2] _−_ 12λ + 8 = 0 megoldása λ = 2 (háromszoros gyök)  2 1 _−1_ x1 0 −2 _−1_ 1  _·_ x2 = 0 2 1 _−1_ _x3_ 0 megoldásai (x1, x2, x3) = α(1, 0, 2) + β(0, 1, 1) alakúak, tehát csak két lineárisan független sajátvektor van. A mátrix mindent (1, −1, 1) többszöröseibe visz, válasszuk ezt az egyik bázisvektornak, egy őse pl. (0, 1, 0), a harmadik lehet (0, 1, 1). Ebből _A = S_ 2 1 0 0 2 0 _S−1,_ 0 0 2 ahol  1 0 0   2 1 _−1_ _._ _−1_ 0 1 _S =_  1 0 0 _−1_ 1 1 _S[−][1]_ =   1 0 1 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">12. feladatsor: Lineáris állandó együtthatós</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">egyenletrendszerek (megoldás)</span></b></p> <p style="top:143.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Mi a Jordan-felbontása az</span></p> <p style="top:171.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:155.4pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:164.0pt;left:141.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:164.0pt;left:166.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:178.5pt;left:136.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:178.5pt;left:161.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> <p style="top:155.4pt;left:177.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:201.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrixnak?</span></p> <p style="top:219.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gyöke</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (kétszeres).</span></p> <p style="top:233.5pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:242.1pt;left:116.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:242.1pt;left:142.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:256.6pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:256.6pt;left:137.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:233.5pt;left:152.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"># "</span></p> <p style="top:242.1pt;left:166.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:256.6pt;left:166.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:233.5pt;left:177.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:249.4pt;left:186.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:233.5pt;left:199.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:242.1pt;left:204.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:256.6pt;left:204.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:233.5pt;left:210.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:279.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">összes megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> többszöröse, ezek a sajátvektorok.</span></p> <p style="top:293.0pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:301.6pt;left:116.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:301.6pt;left:142.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:316.0pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:316.0pt;left:137.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:293.0pt;left:152.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"># "</span></p> <p style="top:301.6pt;left:166.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:316.0pt;left:166.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:293.0pt;left:177.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:308.9pt;left:186.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:293.0pt;left:199.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:301.6pt;left:209.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:316.0pt;left:204.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:293.0pt;left:220.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:338.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egy megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:368.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:352.4pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:361.0pt;left:146.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:361.0pt;left:176.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:151.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:375.5pt;left:171.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:352.4pt;left:186.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:368.3pt;left:194.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:398.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span></p> <p style="top:425.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:409.5pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:418.2pt;left:140.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:418.2pt;left:165.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:432.6pt;left:135.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:432.6pt;left:161.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:409.5pt;left:176.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:425.5pt;left:207.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:409.5pt;left:242.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:418.2pt;left:248.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:418.2pt;left:273.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:432.6pt;left:252.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:432.6pt;left:277.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:409.5pt;left:288.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:425.5pt;left:296.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:455.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Adjuk meg az</span></p> <p style="top:491.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:469.7pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:487.3pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:477.1pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:477.1pt;left:162.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:477.1pt;left:178.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:491.6pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:491.6pt;left:162.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:491.6pt;left:183.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:506.0pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:506.0pt;left:162.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:506.0pt;left:183.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:469.7pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:487.3pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:528.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrix Jordan-felbontását.</span></p> <p style="top:546.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 8 = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (háromszoros gyök)</span></p> <p style="top:561.3pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:578.8pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:568.6pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:568.6pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:568.6pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:583.1pt;left:113.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:583.1pt;left:138.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:583.1pt;left:168.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:597.5pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:597.5pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:597.5pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:561.3pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:578.8pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:561.3pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:578.8pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:568.6pt;left:200.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:583.1pt;left:200.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:597.5pt;left:200.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:561.3pt;left:211.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:578.8pt;left:211.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:561.3pt;left:234.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:578.8pt;left:234.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:568.6pt;left:240.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:583.1pt;left:240.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:597.5pt;left:240.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:561.3pt;left:246.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:578.8pt;left:246.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:620.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldásai</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> β</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakúak, tehát csak két lineárisan független</span></p> <p style="top:634.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátvektor van. A mátrix mindent</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> többszöröseibe visz, válasszuk ezt az egyik</span></p> <p style="top:649.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">bázisvektornak, egy őse pl.</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a harmadik lehet</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ebből</span></p> <p style="top:686.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:664.5pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:682.0pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:671.8pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:671.8pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:671.8pt;left:179.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:686.3pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:686.3pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:686.3pt;left:179.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:700.7pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:700.7pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:700.7pt;left:179.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:664.5pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:682.0pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:722.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span></p> <p style="top:757.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:735.0pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:742.4pt;left:141.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:742.4pt;left:161.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:742.4pt;left:177.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:756.9pt;left:136.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:756.9pt;left:161.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:756.9pt;left:177.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:771.3pt;left:141.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:771.3pt;left:161.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:771.3pt;left:177.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:735.0pt;left:183.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:183.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:757.0pt;left:215.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:735.0pt;left:250.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:250.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:742.4pt;left:261.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:742.4pt;left:282.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:742.4pt;left:302.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:756.9pt;left:261.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:756.9pt;left:282.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:756.9pt;left:298.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:771.3pt;left:257.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:771.3pt;left:282.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:771.3pt;left:302.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:735.0pt;left:313.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:313.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> </div>	page_312.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 12. feladatsor: Lineáris állandó együtthatós egyenletrendszerek (megoldás) 1. Mi a Jordan-felbontása az mátrixnak? .Megoldás. det( A B E összes megoldása 21. ERI egy megoldása 21 9 - AX BA1 6 9 6 9 ahol Adjuk meg az a-[2 [2 mátrix Jordan-felbontását. :] 1] 1 1 1 .Megoldás. det(A — AY — —9 46X" [2 1 -9 FJ [ ÉA [iíJ'LÉiJ győke A — —1 (kétszeres). 123 4-8 — 0 megoldása A — 2 (háromszoros gyök) megoldásai (z1.79.23) — a(1.0.2) 4. 8(0.1.1) alakúak, tehát csak két lincárisan fűggetle sajátvektor van. A mátrix mindent (1, bázisvektornak, 2 AzSsfo o ahol 1.1) többszöröseibe visz, válasszuk ezt az egyik egy őse pl. (0,1.0), a harmadik lehet (0,1.1). Ebből
5. Oldjuk meg az y′ 1 y′ 1 = −2y1 + y2 y′ 2 = −2y2 + y3 y′ 2 = −2y2 + y3 y′ 3 = −2y3 y′ 3 = −2y3 diﬀerenciálegyenlet-rendszert y1(0) = 0, y2(0) = 0, y3(0) = 1 kezdeti feltétel mellett. Megoldás. Az egyenlet y′ = Ay alakú, ahol y = (y1, y2, y3) és   −2 1 0 0 −2 1 0 0 −2 A =  , 2 ami éppen egy 3 × 3 méretű Jordan-blokk, tehát eAx közvetlenül felírható:  1 x x2  eAx = e−2x 0 0 . A kezdeti feltétel y(0) = (0, 0, 1), tehát x2 x2 2 e−2x 2x y(x) = eAxy(0) =  2 xe−2x . e−2x További gyakorló feladatok 6. Határozzuk meg az      4 2 −3 2 0 1 0 0 5 2 −4 2 −1 −2 1 −1  A = mátrix Jordan-féle normálalakját. Megoldás. det(A −λI) a második sor szerint kifejtve: det(A −λI) = (1 −λ) 1 + λ −λ2 −λ3 = λ4 −2λ2 + 1 = (λ2 −1)2 = (λ −1)2(λ + 1)2, tehát a gyökök λ = ±1, mindkettő kétszeres. A λ = 1 sajátértékhez tartozó sajátvektorok:      3 2 −3 2 0 0 0 0 5 2 −5 2 −1 −2 1 −2    x1 x2 x3 x4   x1 x2 x3 x4 =  0 0 0  megoldása alapján (x1, x2, x3, x4) = α(1, 0, 1, 0) + β(0, 1, 0, −1) alakúak, tehát két 1 × 1 méretű Jordan-blokk van. A λ = −1 sajátértékhez tartozó sajátvektorok:   5 2 −3 0 2 0 5 2 −3 −1 −2 1  x x    x1 x2 x3 x4 2 3 =   0 0 0	5. Oldjuk meg az _y1[′]_ [=][ −][2][y][1] [+][ y][2] _y2[′]_ [=][ −][2][y][2] [+][ y][3] _y3[′]_ [=][ −][2][y][3] differenciálegyenlet-rendszert y1(0) = 0, y2(0) = 0, y3(0) = 1 kezdeti feltétel mellett. _Megoldás. Az egyenlet y[′]_ = Ay alakú, ahol y = (y1, y2, y3) és _A =_ −2 1 0   0 _−2_ 1  _,_ 0 0 _−2_ ami éppen egy 3 × 3 méretű Jordan-blokk, tehát e[Ax] közvetlenül felírható: _e[Ax]_ = e[−][2][x] 1 _x_ _x[2]_ 2 0 1 _x_ 0 0 1   _._ A kezdeti feltétel y(0) = (0, 0, 1), tehát _x[2]_ 2 _[e][−][2][x]_ _xe[−][2][x]_ _e[−][2][x]_   _._ y(x) = e[Ax]y(0) =   ## További gyakorló feladatok 6. Határozzuk meg az _A =_  4 2 _−3_ 2  0 1 0 0 5 2 _−4_ 2   _−1_ _−2_ 1 _−1_ mátrix Jordan-féle normálalakját. _Megoldás. det(A −_ _λI) a második sor szerint kifejtve:_ det(A − _λI) = (1 −_ _λ)_ �1 + λ − _λ[2]_ _−_ _λ[3][�]_ = λ[4] _−_ 2λ[2] + 1 = (λ[2] _−_ 1)[2] = (λ − 1)[2](λ + 1)[2], tehát a gyökök λ = ±1, mindkettő kétszeres. A λ = 1 sajátértékhez tartozó sajátvektorok:  3 2 _−3_ 2  x1 0 0 0 0 0 _x2_ = 0 5 2 _−5_ 2 _x3_ 0       _−1_ _−2_ 1 _−2_ _x4_ 0 megoldása alapján (x1, x2, x3, x4) = α(1, 0, 1, 0) + β(0, 1, 0, −1) alakúak, tehát két 1 × 1 méretű Jordan-blokk van. A λ = −1 sajátértékhez tartozó sajátvektorok:  5 2 _−3_ 2 x1 0 0 2 0 0 _x2_ = 0 5 2 _−3_ 2 _x3_ 0       _−1_ _−2_ 1 0 _x4_ 0 -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Oldjuk meg az</span></p> <p style="top:81.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:87.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:98.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:104.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:116.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:122.2pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:138.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:155.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenlet</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakú, ahol</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></p> <p style="top:192.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:170.2pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:187.8pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:177.6pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:177.6pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:177.6pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:192.1pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:192.1pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:192.1pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:206.5pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:206.5pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:206.5pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:170.2pt;left:203.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:187.8pt;left:203.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:229.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ami éppen egy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> méretű Jordan-blokk, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ax</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">közvetlenül felírható:</span></p> <p style="top:265.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ax</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:243.7pt;left:162.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:261.2pt;left:162.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:251.7pt;left:169.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:251.7pt;left:185.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:250.2pt;left:203.0pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:259.0pt;left:205.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:266.1pt;left:169.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:266.1pt;left:185.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:266.1pt;left:204.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:280.6pt;left:169.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:280.6pt;left:185.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:280.6pt;left:204.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:243.7pt;left:213.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:261.2pt;left:213.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:302.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdeti feltétel</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:339.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ax</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span></p> <p style="top:317.6pt;left:200.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:335.1pt;left:200.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:324.1pt;left:207.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:332.9pt;left:210.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:340.1pt;left:208.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:354.5pt;left:212.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:317.6pt;left:239.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:335.1pt;left:239.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:383.8pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:407.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Határozzuk meg az</span></p> <p style="top:450.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:422.9pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:440.4pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:429.0pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:429.0pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:429.0pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:429.0pt;left:217.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:443.5pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:443.5pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:443.5pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:443.5pt;left:217.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:457.9pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:457.9pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:457.9pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:457.9pt;left:217.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:472.4pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:472.4pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:472.4pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:472.4pt;left:212.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:422.9pt;left:228.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:440.4pt;left:228.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:493.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrix Jordan-féle normálalakját.</span></p> <p style="top:511.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a második sor szerint kifejtve:</span></p> <p style="top:534.8pt;left:106.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:524.9pt;left:222.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:534.8pt;left:228.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></sup></p> <p style="top:554.9pt;left:171.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span></p> <p style="top:572.4pt;left:171.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:589.8pt;left:171.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:611.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a gyökök</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, mindkettő kétszeres. A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sajátértékhez tartozó sajátvektorok:</span></p> <p style="top:626.9pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:644.5pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:633.0pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:633.0pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:633.0pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:633.0pt;left:193.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:647.5pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:647.5pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:647.5pt;left:168.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:647.5pt;left:193.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:661.9pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:661.9pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:661.9pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:661.9pt;left:193.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:676.4pt;left:113.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:676.4pt;left:138.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:676.4pt;left:168.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:676.4pt;left:188.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:626.9pt;left:203.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:644.5pt;left:203.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:626.9pt;left:212.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:644.5pt;left:212.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:633.0pt;left:218.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:647.5pt;left:218.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:661.9pt;left:218.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:676.4pt;left:218.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></p> <p style="top:626.9pt;left:230.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:644.5pt;left:230.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:626.9pt;left:252.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:644.5pt;left:252.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:633.0pt;left:259.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:647.5pt;left:259.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:661.9pt;left:259.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:676.4pt;left:259.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:626.9pt;left:265.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:644.5pt;left:265.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:698.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása alapján</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0) +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> β</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakúak, tehát két</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></p> <p style="top:712.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">méretű Jordan-blokk van. A</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sajátértékhez tartozó sajátvektorok:</span></p> <p style="top:727.8pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.3pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:733.9pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:733.9pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:733.9pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:733.9pt;left:188.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:748.4pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:748.4pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:748.4pt;left:168.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:748.4pt;left:188.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:762.8pt;left:117.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:762.8pt;left:142.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:762.8pt;left:163.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:762.8pt;left:188.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:777.3pt;left:113.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:777.3pt;left:138.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:777.3pt;left:168.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:777.3pt;left:188.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:727.8pt;left:194.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.3pt;left:194.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:727.8pt;left:202.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.3pt;left:202.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:733.9pt;left:209.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:748.4pt;left:209.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:762.8pt;left:209.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:777.3pt;left:209.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></p> <p style="top:727.8pt;left:220.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.3pt;left:220.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:727.8pt;left:243.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.3pt;left:243.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:733.9pt;left:250.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:748.4pt;left:250.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:762.8pt;left:250.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:777.3pt;left:250.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:727.8pt;left:255.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:745.3pt;left:255.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> </div>	page_314.png	5. Oldjuk meg az W$ tn 2n t 29 95(0) — 1 kezdeti feltétel mellett Gn.y2,n) és dszert y1(0) — 0. yz(0) t y/ — Ay alakú, ahol y differenciálegyer "Megoldás. Az egyen blokk, tehát e" közvetlenül felírható: v - ey0- "További gyakorló feladatok 3 2 Dr 4 2 l 2 1 A le normálalakját. mátrix Jordat . Megoldás. det( A — AZ) a második sor szerint kifejtve N3 241 1 11. h 2 1 2] [ megoldása alapján (71.21.23.24) — a(1.0.1.0) 4- 8(0.1.0,—1) alakúak, t. Jordan-blokk van. A A — —1 sajátért ó sajátvektorok: É 2 s3É-8 1 -2 1 ollal ól dot(A — aT! sajátértékhez tartozó sajátvektorok: két 1561
8. Oldjuk meg az y′ = y −1 3 diﬀerenciálegyenlet-rendszert y(0) = (3, 2) kezdeti feltétel mellett. Megoldás. Legyen A az együtthatómátrix, a sajátértékek det(A−λI) = λ2−4λ+4 = (λ−2)2 gyökei, tehát a 2 kétszeres multiplicitással. A sajátvektorok a " −1 −1 # " x1 x2 egyenletrendszer nemtriviális megoldásai, ezek mind (1, 1) többszörösei, tehát a geometriai multiplicitás 1. Legyen (1, 1) az egyik bázisvektor, ennek egy őse az A−2I leképezés szerint (0, 1), ezt választhatjuk másiknak. Vezessük be az S = S−1 = −1 1 mátrixot, ezzel A = S S−1. A kezdetiérték-probléma megoldása y(x) = eAxy(0) = S " e2x xe2x 0 e2x S−1 " 3e2x −xe2x 2e2x −xe2x 9. Oldjuk meg az     10 −19 13 1 −1 1 −9 18 −12 y′ =  y 2 diﬀerenciálegyenlet-rendszert y(0) = (1, 0, 1) kezdeti feltétel mellett. Megoldás. Legyen A az együtthatómátrix, a karakterisztikus polinomja det(A−λI) = −λ3− 3λ2 = −λ2(λ + 3), tehát 0 kétszeres, −3 egyszeres gyök. A 0 sajátértékhez csak (−2, 1, 3) többszörösei a sajátvektorok (a geometriai multiplicitás 1), tehát 2 × 2 méretű Jordanblokk tartozik hozzá. Legyen az első bázisvektor (−2, 1, 3), ennek egy őse (az A −0I = A leképezés szerint) a második, például (1, 2, 2), a harmadik pedig a 3 sajátértékhez tartozó egyik sajátvektor, például (1, 0, −1). Vezessük be az     −2 1 1 1 2 0 3 2 −1   S−1 =     −2 3 −2 1 −1 1 −4 7 −5 S =    5 mátrixot, ezzel S−1. 0 −3 A = S 0 0 A kezdetiérték-probléma megoldása  1 1 x 0  10 −4x −9e−3x  y(x) = eAxy(x) = S 0 0 S−1 0 e−3x  0 =  2x −8 + 6x + 9e−3x  2x −8 + 6x + 9e−3x 	8. Oldjuk meg az y[′] = � 1 1� y _−1_ 3 differenciálegyenlet-rendszert y(0) = (3, 2) kezdeti feltétel mellett. _Megoldás. Legyen A az együtthatómátrix, a sajátértékek det(A−λI) = λ[2]−4λ+4 = (λ−2)[2]_ gyökei, tehát a 2 kétszeres multiplicitással. A sajátvektorok a �−1 1��x1� = �0� _−1_ 1 _x2_ 0 egyenletrendszer nemtriviális megoldásai, ezek mind (1, 1) többszörösei, tehát a geometriai multiplicitás 1. Legyen (1, 1) az egyik bázisvektor, ennek egy őse az A−2I leképezés szerint (0, 1), ezt választhatjuk másiknak. Vezessük be az � 1 0� _−1_ 1 _S =_ �1 0� _S[−][1]_ = 1 1 mátrixot, ezzel _A = S_ �2 1� _S[−][1]._ 0 2 A kezdetiérték-probléma megoldása �3e[2][x] _−_ _xe[2][x]_ 2e[2][x] _−_ _xe[2][x]_ � _._ y(x) = e[Ax]y(0) = S 9. Oldjuk meg az �e[2][x] _xe[2][x]�_ �3� _S[−][1]_ = 0 _e[2][x]_ 2 y[′] =  10 _−19_ 13   1 _−1_ 1  y _−9_ 18 _−12_ differenciálegyenlet-rendszert y(0) = (1, 0, 1) kezdeti feltétel mellett. _Megoldás. Legyen A az együtthatómátrix, a karakterisztikus polinomja det(A−λI) = −λ[3]−_ 3λ[2] = −λ[2](λ + 3), tehát 0 kétszeres, −3 egyszeres gyök. A 0 sajátértékhez csak (−2, 1, 3) többszörösei a sajátvektorok (a geometriai multiplicitás 1), tehát 2 × 2 méretű Jordanblokk tartozik hozzá. Legyen az első bázisvektor (−2, 1, 3), ennek egy őse (az A − 0I = A leképezés szerint) a második, például (1, 2, 2), a harmadik pedig a 3 sajátértékhez tartozó egyik sajátvektor, például (1, 0, −1). Vezessük be az −2 3 _−2_  1 _−1_ 1  _−4_ 7 _−5_ _S =_ −2 1 1  1 2 0 _S[−][1]_ =   3 2 _−1_ mátrixot, ezzel _A = S_ 0 1 0  0 0 0  _S−1._ 0 0 _−3_ A kezdetiérték-probléma megoldása   _S−1_ 1 0 = 1  10 − 4x − 9e[−][3][x]  2x _−8 + 6x + 9e[−][3][x]_   y(x) = e[Ax]y(x) = S 1 _x_ 0 0 1 0 0 0 _e[−][3][x]_ -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Oldjuk meg az</span></p> <p style="top:88.3pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:72.4pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:81.0pt;left:142.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:81.0pt;left:163.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:95.4pt;left:137.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:95.4pt;left:163.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:72.4pt;left:168.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:88.3pt;left:176.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:118.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:135.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az együtthatómátrix, a sajátértékek</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+4 = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:150.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">gyökei, tehát a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kétszeres multiplicitással. A sajátvektorok a</span></p> <p style="top:163.6pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:172.3pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:172.3pt;left:137.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:186.7pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:186.7pt;left:137.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:163.6pt;left:143.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"># "</span></p> <p style="top:172.3pt;left:156.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:186.7pt;left:156.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:163.6pt;left:168.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:179.6pt;left:177.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:163.6pt;left:189.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:172.3pt;left:195.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:186.7pt;left:195.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:163.6pt;left:201.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:209.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszer nemtriviális megoldásai, ezek mind</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> többszörösei, tehát a geometriai</span></p> <p style="top:223.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">multiplicitás</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Legyen</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az egyik bázisvektor, ennek egy őse az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> leképezés szerint</span></p> <p style="top:238.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ezt választhatjuk másiknak. Vezessük be az</span></p> <p style="top:268.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:252.3pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:260.9pt;left:135.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:260.9pt;left:151.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:275.4pt;left:135.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:275.4pt;left:151.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:252.3pt;left:157.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:268.3pt;left:188.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:252.3pt;left:223.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:260.9pt;left:234.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:260.9pt;left:254.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:275.4pt;left:229.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:275.4pt;left:254.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:252.3pt;left:260.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:297.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrixot, ezzel</span></p> <p style="top:326.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:310.7pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:319.3pt;left:146.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:319.3pt;left:162.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:333.8pt;left:146.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:333.8pt;left:162.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:310.7pt;left:168.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:326.6pt;left:176.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:355.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdetiérték-probléma megoldása</span></p> <p style="top:385.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ax</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:369.1pt;left:209.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:377.7pt;left:215.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:377.7pt;left:240.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:392.1pt;left:220.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:392.1pt;left:243.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:369.1pt;left:262.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:385.0pt;left:269.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:369.1pt;left:291.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:377.7pt;left:296.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:392.1pt;left:296.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:369.1pt;left:302.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:385.0pt;left:311.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:369.1pt;left:324.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:377.7pt;left:330.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:392.1pt;left:330.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:369.1pt;left:387.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:385.0pt;left:394.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:414.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Oldjuk meg az</span></p> <p style="top:450.1pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:428.2pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:445.7pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:435.6pt;left:140.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span></p> <p style="top:435.6pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">19</span></p> <p style="top:435.6pt;left:199.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">13</span></p> <p style="top:450.0pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:450.0pt;left:166.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:450.0pt;left:202.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:464.5pt;left:138.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:464.5pt;left:168.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">18</span></p> <p style="top:464.5pt;left:194.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12</span></p> <p style="top:428.2pt;left:215.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:445.7pt;left:215.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:486.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:504.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az együtthatómátrix, a karakterisztikus polinomja</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></p> <p style="top:518.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kétszeres,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyszeres gyök. A</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sajátértékhez csak</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span></p> <p style="top:533.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">többszörösei a sajátvektorok (a geometriai multiplicitás</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> méretű Jordan-</span></p> <p style="top:547.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">blokk tartozik hozzá. Legyen az első bázisvektor</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ennek egy őse (az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i></p> <p style="top:562.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">leképezés szerint) a második, például</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a harmadik pedig a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sajátértékhez tartozó</span></p> <p style="top:576.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyik sajátvektor, például</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Vezessük be az</span></p> <p style="top:613.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:591.4pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:608.9pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:598.7pt;left:136.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:598.7pt;left:161.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:598.7pt;left:182.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:613.2pt;left:141.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:613.2pt;left:161.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:613.2pt;left:182.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:627.6pt;left:141.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:627.6pt;left:161.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:627.6pt;left:177.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:591.4pt;left:192.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:608.9pt;left:192.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:613.3pt;left:224.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:591.4pt;left:259.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:608.9pt;left:259.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:598.7pt;left:266.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:598.7pt;left:296.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:598.7pt;left:316.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:613.2pt;left:271.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:613.2pt;left:291.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:613.2pt;left:321.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:627.6pt;left:266.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:627.6pt;left:296.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> <p style="top:627.6pt;left:316.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:591.4pt;left:331.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:608.9pt;left:331.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:649.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrixot, ezzel</span></p> <p style="top:685.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:663.2pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:680.7pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:670.6pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:670.6pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:670.6pt;left:183.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:685.0pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:685.0pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:685.0pt;left:183.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:699.5pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:699.5pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:699.5pt;left:179.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:663.2pt;left:194.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:680.7pt;left:194.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:721.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdetiérték-probléma megoldása</span></p> <p style="top:757.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ax</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:735.0pt;left:210.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:210.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:742.4pt;left:217.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:742.4pt;left:233.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:742.4pt;left:257.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:756.9pt;left:217.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:756.9pt;left:233.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:756.9pt;left:257.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:771.3pt;left:217.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:771.3pt;left:233.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:771.3pt;left:249.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:735.0pt;left:271.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:271.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:735.0pt;left:301.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:301.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:742.4pt;left:307.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:756.9pt;left:307.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:771.3pt;left:307.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:735.0pt;left:313.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:313.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:735.0pt;left:335.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:335.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:742.4pt;left:344.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:756.9pt;left:378.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:771.3pt;left:342.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8 + 6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 9</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:735.0pt;left:426.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:752.6pt;left:426.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_316.png	8. Oldjuk meg az ala a) V3 differenciálegyer dszert y(0) — (3,2) kezdeti feltétel mellett ek det(A-—A1! a 2 kétszeres multiplicítással. A sajátvektorok a E TÉLŰ egyenletrendszer nemtriviális megoldásai, ezek mind (1,1) többszörösei, tehát a e multiplicitás 1. Legyen (1.1) az egyik bázisvektor, ennek egy őse az A — 21/ leké; (0. 1). ezt választhatjuk másiknak. Vi [ ÉJ mátrixot, ezzel A kezdetiértél . Megoldás. Legyen A az együtthatómátrix, a sajátértél gyökei, t X-- ssük be az problé vo-évo-s[g 15 [-fÉE] sa megoldása 9. Oldjuk meg az 10 -19 13 Y2ll - 1ly 9 18 — differenciálegyer dszert y(0) — (1.0.1) kezdeti feltétel mellett. " Megoldás. Legyen A az együtthatómátrix, a karakterisztikus polinomja det(A—AT) — — 332 — —J2(X 4 3). tehát 0 kétszeres, —3 egyszeres győk. A 0 sajátértékhez csak (-2.1.3) többszörősei a sajátvektorok (a geometriai multiplicitás 1). tehát 2 x 2 méretű Jordat blokk tartozik hozzá. Legyen az első bázisvektor (-—2,1.3), ennek egy őse (az A — 07 — A teképezés szerint) a második, például (1,2.2) a harmadik pedig a 3 sajátértékhez tartozó Idául (1.0,—1). Vezessük be az 21 1 2 3 12 0] s--fi - 3 2 -] 4 7 mátrixot, ezzel o1 0 sjo 0 ols-. o 0 -3] A kezdetiértél 12 0 1) [10-4r—989- v -eyezesfó 1 0 S- - 22 po] ] [sz6r90s egyik sajátvektor, p
10. Oldjuk meg az     5 −3 4 6 −3 3 −1 1 −2 y′ =  y diﬀerenciálegyenlet-rendszert y(0) = (1, 1, −1) kezdeti feltétel mellett. Megoldás. Legyen A az együtthatómátrix, a sajárértékek det(A −λI) = −λ3 gyökei, tehát a 0 háromszoros gyök. A hozzá tartozó sajátvektorok (1, 3, 1) többszörösei (a geometriai multiplicitás 1), tehát egy 3 × 3 Jordan-blokk van. Válasszuk az első bázisvektornak ezt a vektort, a második legyen ennek egy őse (az A−0I = A leképezés szerint), például (2, 3, 0), az utolsó pedig ennek egy őse, például (1, 1, 0). Vezessük be az S =    S−1 =   −1 1 −2 3 −2 3  mátrixot, ezzel A = S 0 0 S−1. 1 0 A kezdetiérték-probléma megoldása x2 1 −2x −x2  y(x) = eAxy(x) = S 0 0 S−1    −1 =    −1 + 2x −x2 1 −3x2 . 11. ∗Határozzuk meg az y′′ +Ω2y = 0 diﬀerenciálegyenlet-rendszer általános megoldását, ha Ω n × n méretű szimmetrikus pozitív deﬁnit mátrix. Mutassuk meg, hogy minden megoldás korlátos. Megoldás. Az egyenletrendszer másodrendű, ezt átírhatjuk 2n egyenletből álló elsőrendű egyenletrendszerré. Ennek mátrixa blokk-mátrix alakban írható fel legegyszerűbben: A = . −Ω2 0 Ennek kell megkeresni a sajátértékeit és sajátvektorait. A karakterisztikus polinom det(A −λI) = −λI I −Ω2 −λI −λ2I −Ω2 −λI I 0 −λI −λ2I −Ω2 = det(Ω2 + λ2I), = (−1)n tehát a sajátértékeket úgy kaphatjuk meg, hogy Ω2 sajátértékeinek ellentettjeiből négyzetgyököt vonunk (mindegyiknek 2 négyzetgyöke van). Ezek mind tisztán képzetesek, tehát a megoldások korlátosak lesznek. A sajátvektorokat szintén blokk-alakban érdemes keresni, " # " # " # " # x x y 0 I λ −Ω2 −Ω2x	10. Oldjuk meg az y[′] =  5 _−3_ 4   6 _−3_ 3  y _−1_ 1 _−2_ differenciálegyenlet-rendszert y(0) = (1, 1, −1) kezdeti feltétel mellett. _Megoldás. Legyen A az együtthatómátrix, a sajárértékek det(A −_ _λI) = −λ[3]_ gyökei, tehát a 0 háromszoros gyök. A hozzá tartozó sajátvektorok (1, 3, 1) többszörösei (a geometriai multiplicitás 1), tehát egy 3 × 3 Jordan-blokk van. Válasszuk az első bázisvektornak ezt a vektort, a második legyen ennek egy őse (az A _−_ 0I = A leképezés szerint), például (2, 3, 0), az utolsó pedig ennek egy őse, például (1, 1, 0). Vezessük be az 1 2 1 _S =_ 3 3 1 _S[−][1]_ = 1 0 0  0 0 1  −1 1 _−2_ 3 _−2_ 3 mátrixot, ezzel _A = S_ 0 1 0 0 0 1 _S−1._ 0 0 0 A kezdetiérték-probléma megoldása  1   1  = _−1_  1 − 2x − _x[2]_ 1 − 3x[2]  _−1 + 2x −_ _x[2]_   _._ y(x) = e[Ax]y(x) = S 1 _x_ _x[2]_ 2 0 1 _x_ 0 0 1   _S−1_ 11. ∗ Határozzuk meg az y[′′] +Ω[2]y = 0 differenciálegyenlet-rendszer általános megoldását, ha Ω _n × n méretű szimmetrikus pozitív definit mátrix. Mutassuk meg, hogy minden megoldás_ korlátos. _Megoldás. Az egyenletrendszer másodrendű, ezt átírhatjuk 2n egyenletből álló elsőrendű_ egyenletrendszerré. Ennek mátrixa blokk-mátrix alakban írható fel legegyszerűbben: � 0 _I_ _A =_ _−Ω[2]_ 0 � _._ Ennek kell megkeresni a sajátértékeit és sajátvektorait. A karakterisztikus polinom _−λI_ _I_ 0 _I_ det(A − _λI) =_ = _−Ω[2]_ _−λI_ _−λ[2]I −_ Ω[2] _−λI_ �� = (−1)[n] _I_ 0 = det(Ω2 + λ2I), _−λI_ _−λ[2]I −_ Ω[2] �� tehát a sajátértékeket úgy kaphatjuk meg, hogy Ω[2] sajátértékeinek ellentettjeiből négyzetgyököt vonunk (mindegyiknek 2 négyzetgyöke van). Ezek mind tisztán képzetesek, tehát a megoldások korlátosak lesznek. A sajátvektorokat szintén blokk-alakban érdemes keresni, � 0 _I_ _−Ω[2]_ 0 ��x� = y � y _−Ω[2]x_ � _λ_ �x� = y -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">10. Oldjuk meg az</span></p> <p style="top:99.0pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:77.1pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:94.6pt;left:132.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:84.5pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:84.5pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:84.5pt;left:193.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:98.9pt;left:143.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> <p style="top:98.9pt;left:163.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:98.9pt;left:193.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:113.4pt;left:138.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:113.4pt;left:168.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:113.4pt;left:189.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:77.1pt;left:204.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:94.6pt;left:204.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:139.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszert</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett.</span></p> <p style="top:159.0pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az együtthatómátrix, a sajárértékek</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">gyökei, tehát</span></p> <p style="top:173.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> háromszoros gyök. A hozzá tartozó sajátvektorok</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> többszörösei (a geometriai</span></p> <p style="top:187.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">multiplicitás</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), tehát egy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Jordan-blokk van. Válasszuk az első bázisvektornak ezt a</span></p> <p style="top:202.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vektort, a második legyen ennek egy őse (az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> leképezés szerint), például</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:216.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az utolsó pedig ennek egy őse, például</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Vezessük be az</span></p> <p style="top:257.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:235.5pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:253.0pt;left:130.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:242.9pt;left:136.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:242.9pt;left:152.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:242.9pt;left:168.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:257.3pt;left:136.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:257.3pt;left:152.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:257.3pt;left:168.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:271.8pt;left:136.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:271.8pt;left:152.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:271.8pt;left:168.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:235.5pt;left:174.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:253.0pt;left:174.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:257.4pt;left:206.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:235.5pt;left:241.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:253.0pt;left:241.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:242.9pt;left:252.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:242.9pt;left:277.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:242.9pt;left:302.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:257.3pt;left:247.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:257.3pt;left:277.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:257.3pt;left:298.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:271.8pt;left:252.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:271.8pt;left:273.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:271.8pt;left:302.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:235.5pt;left:313.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:253.0pt;left:313.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:297.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrixot, ezzel</span></p> <p style="top:337.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:315.3pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:332.8pt;left:140.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:322.7pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:322.7pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:322.7pt;left:179.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:337.1pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:337.1pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:337.1pt;left:179.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:351.6pt;left:147.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:351.6pt;left:163.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:351.6pt;left:179.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:315.3pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:332.8pt;left:185.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:377.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A kezdetiérték-probléma megoldása</span></p> <p style="top:417.7pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ax</span></i></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> S</span></i></p> <p style="top:395.7pt;left:210.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:413.3pt;left:210.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:403.7pt;left:217.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:403.7pt;left:233.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:402.2pt;left:250.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:411.0pt;left:253.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:418.2pt;left:217.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:418.2pt;left:233.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:418.2pt;left:252.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></p> <p style="top:432.6pt;left:217.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:432.6pt;left:233.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:432.6pt;left:252.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:395.7pt;left:261.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:413.3pt;left:261.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">S</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:395.7pt;left:290.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:413.3pt;left:290.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:403.1pt;left:302.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:417.6pt;left:302.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:432.0pt;left:297.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:395.7pt;left:312.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:413.3pt;left:312.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:395.7pt;left:335.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:413.3pt;left:335.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:403.1pt;left:346.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:417.6pt;left:356.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:432.0pt;left:341.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 + 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:395.7pt;left:409.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:413.3pt;left:409.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:459.4pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">11.</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∗</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Határozzuk meg az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenlet-rendszer általános megoldását, ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span></p> <p style="top:473.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> méretű szimmetrikus pozitív deﬁnit mátrix. Mutassuk meg, hogy minden megoldás</span></p> <p style="top:488.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">korlátos.</span></p> <p style="top:507.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenletrendszer másodrendű, ezt átírhatjuk</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenletből álló elsőrendű</span></p> <p style="top:522.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszerré. Ennek mátrixa blokk-mátrix alakban írható fel legegyszerűbben:</span></p> <p style="top:555.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:539.4pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:548.0pt;left:145.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:548.0pt;left:169.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:562.5pt;left:136.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:562.5pt;left:169.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:539.4pt;left:175.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:555.4pt;left:183.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:588.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ennek kell megkeresni a sajátértékeit és sajátvektorait. A karakterisztikus polinom</span></p> <p style="top:621.7pt;left:108.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:605.4pt;left:185.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:614.4pt;left:189.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:614.4pt;left:229.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:628.8pt;left:189.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:628.8pt;left:221.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:605.4pt;left:243.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:605.4pt;left:262.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:614.4pt;left:290.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:614.4pt;left:339.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:628.8pt;left:266.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:628.8pt;left:331.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:605.4pt;left:353.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:655.6pt;left:173.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup></p> <p style="top:639.2pt;left:217.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:648.3pt;left:229.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:648.3pt;left:277.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:662.7pt;left:221.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:662.7pt;left:253.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:639.2pt;left:308.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = det(Ω</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:690.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát a sajátértékeket úgy kaphatjuk meg, hogy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátértékeinek ellentettjeiből négyzet-</span></p> <p style="top:704.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">gyököt vonunk (mindegyiknek</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> négyzetgyöke van). Ezek mind tisztán képzetesek, tehát a</span></p> <p style="top:718.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldások korlátosak lesznek.</span></p> <p style="top:733.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A sajátvektorokat szintén blokk-alakban érdemes keresni,</span></p> <p style="top:766.5pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></p> <p style="top:750.6pt;left:115.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:759.2pt;left:121.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">x</span></b></p> <p style="top:773.6pt;left:121.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:750.6pt;left:128.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:766.5pt;left:137.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:750.6pt;left:149.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:759.2pt;left:164.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:759.2pt;left:188.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:773.6pt;left:155.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:773.6pt;left:188.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:750.6pt;left:194.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"># "</span></p> <p style="top:759.2pt;left:207.8pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">x</span></b></p> <p style="top:773.6pt;left:207.9pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:750.6pt;left:215.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:766.5pt;left:224.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:750.6pt;left:236.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:759.2pt;left:253.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:773.6pt;left:242.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">x</span></b></p> <p style="top:750.6pt;left:272.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> </div>	page_317.png	10. Oldjuk meg az differenciálegyer dszert y(0) — (1.1,—1) kezdeti feltétel mellett .Megoldás. Legyen A az együttható a0 multiplicitás 1). tehát egy 3 x 3 Jordan-blokk van. Válasszuk az első bázisvektornak e vektort, a második legyen ennek egy őse (az A—0/ — A leképezés szerint), például (2.3.0). az utolsó ped egy őse, például (1.1,0). Vezessük be az 12 3 3 10 mátrixot, ezzel sátrix, a sajárértékek det(A — AT) — —9? győkei, tehát o 0 0 A kezdetiértél yo-eyesb 3 l1 [E] sA b ss m x n méretű szimmetrikus pozitív definít mátrix. Mutassuk meg, hogy minden megoldás korlátos. .Megoldás. Az egyenletrendszer másodrendü, ezt átírhatjuk 2n egyenletből álló elsőrendüű egyenletrendszerré. Etnek mátrixa blokk-mátrix alakban írható fel legegyszerűbbi h Ennek kell megkeresni a sajátértékeit és sajátvektorait. A karakterisztikus polinom M OI [] 1 M ÉT dot(A — aT! C er de(P 4 32D), tékeket úgy kaphatjuk meg, hogy £2? sajátértékeinek ellentettjeiből négyzet- győköt vonunk (mindegyiknek 2 négyzetgyőke van). Ezek mind tisztán képzetesek, tehát a megoldások korlátosak lesznek. A sajátvektorokat szintén blokk-alakban érdemes keresni, AA -[ -] 6
akkor teljesül, ha λx = y és λy = −Ω2x, tehát −λ2y = Ω2y. Ez azt jelenti, hogy y (és x is) Ω2 sajátvektora −λ2 sajátértékkel. Ha választunk egy bázist Ωsajátvektoraiból, akkor így képezhetünk A sajátvektoraiból álló bázist is, amivel az általános megoldás a szokásos módon felírható. Legyenek Ωlineárisan független sajátvektorai v1, . . . , vn, a hozzájuk tartozó sajátértékek ω1, . . . , ωn. Ekkor az általános megoldás y(x) = X (Ci cos(ωix)vi + Di sin(ωix)vi). i=1 Egy másik lehetőség: az elsőrendű esethez hasonlóan itt is gondolhatunk az n = 1 speciális esetre, ekkor tudjuk, hogy cos és sin adja a megoldást. Hatványsorral ezek a függvények is értelmezhetőek mátrix argumentumra (ehhez még az sem szükséges, hogy Ωpozitív deﬁnit vagy szimmetrikus legyen!), az általános megoldás y(x) = cos(Ωx)y0+sin(Ωx)Ω−1y′ 0. Persze ennek kiszámításához szintén Ωsajátértékeit és sajátvektorait érdemes megkeresni. Egy másik lehetőség: az elsőrendű esethez hasonlóan itt is gondolhatunk az n = 1 speciális esetre, ekkor tudjuk, hogy cos és sin adja a megoldást. Hatványsorral ezek a függvények is értelmezhetőek mátrix argumentumra (ehhez még az sem szükséges, hogy Ωpozitív deﬁnit vagy szimmetrikus legyen!), az általános megoldás y(x) = cos(Ωx)y0+sin(Ωx)Ω−1y′ 0. 12. ∗Legyenek M, C, K n×n-es mátrixok, M invertálható, és tekintsük az My′′+Cy′+Ky = 0 másodrendű diﬀerenciálegyenlet-rendszert. a) Írjuk át elsőrendű egyenletrendszerré az y, y′ komponenseit tartalmazó (2n elemű) vek- torértékű függvényre nézve. b) Mutassuk meg, hogy az így kapott elsőrendű állandó együtthatós diﬀerenciálegyenlet- rendszer mátrixának sajátértékei a det(λ2M + λC + K) = 0 egyenlet gyökei. (Az ilyen típusú egyenletek neve kvadratikus sajátérték-probléma.) c) Tegyük fel, hogy M, C és K mindegyike pozitív deﬁnit. Mutassuk meg, hogy ekkor minden sajátérték valós része negatív. (Ebből következik, hogy minden megoldás 0-hoz tart amint x →∞.) Megoldás. a) Az elsőrendű egyenletrendszer blokk-mátrix alakban így írható fel: " #′ " # " # y y 0 I y′ −M −1K −M −1C {z A . y′ b) A sajátértékek a det(A−λI) polinom gyökei. Sor- és oszlopműveletekkel a determináns így egyszerűsíthető: −λI I −M −1K −M −1C −λI 0 I −M −1K −λM −1C −λ2I −M −1C −λI det(A −λI) = = (−1)n M −1K + λM −1C + λ2I M −1C + λI = M −1C + λI M −1K + λM −1C + λ2I = det(M −1K + λM −1C + λ2I) = 1 det M det(λ2M + λC + K). Az első lépésben az i. oszlophoz hozzáadtuk az i + n. oszlop λ-szorosát (i = 1, . . . , n), a másodikban az utolsó n sort megszoroztuk −1-gyel, ezután az i. és i + n. oszlopot felcseréltük (i = 1, . . . , n), végül az első n sor szerint kifejtettük a determinánst, és felhasználtuk a determináns multiplikativitását.	akkor teljesül, ha λx = y és λy = −Ω[2]x, tehát −λ[2]y = Ω[2]y. Ez azt jelenti, hogy y (és x is) Ω[2] sajátvektora −λ[2] sajátértékkel. Ha választunk egy bázist Ωsajátvektoraiból, akkor így képezhetünk A sajátvektoraiból álló bázist is, amivel az általános megoldás a szokásos módon felírható. Legyenek Ωlineárisan független sajátvektorai v1, . . ., vn, a hozzájuk tartozó sajátértékek ω1, . . ., ωn. Ekkor az általános megoldás y(x) = _n_ �(Ci cos(ωix)vi + Di sin(ωix)vi). _i=1_ Egy másik lehetőség: az elsőrendű esethez hasonlóan itt is gondolhatunk az n = 1 speciális esetre, ekkor tudjuk, hogy cos és sin adja a megoldást. Hatványsorral ezek a függvények is értelmezhetőek mátrix argumentumra (ehhez még az sem szükséges, hogy Ωpozitív definit vagy szimmetrikus legyen!), az általános megoldás y(x) = cos(Ωx)y0+sin(Ωx)Ω[−][1]y0[′] [.] Persze ennek kiszámításához szintén Ωsajátértékeit és sajátvektorait érdemes megkeresni. 12. ∗ Legyenek M, C, K n×n-es mátrixok, M invertálható, és tekintsük az M y[′′]+Cy[′]+Ky = 0 másodrendű differenciálegyenlet-rendszert. a) Írjuk át elsőrendű egyenletrendszerré az y, y[′] komponenseit tartalmazó (2n elemű) vektorértékű függvényre nézve. b) Mutassuk meg, hogy az így kapott elsőrendű állandó együtthatós differenciálegyenletrendszer mátrixának sajátértékei a det(λ[2]M + λC + K) = 0 egyenlet gyökei. (Az ilyen típusú egyenletek neve kvadratikus sajátérték-probléma.) c) Tegyük fel, hogy M, C és K mindegyike pozitív definit. Mutassuk meg, hogy ekkor minden sajátérték valós része negatív. (Ebből következik, hogy minden megoldás 0-hoz tart amint x →∞.) _Megoldás._ a) Az elsőrendű egyenletrendszer blokk-mátrix alakban így írható fel: � 0 _I_ _−M_ _[−][1]K_ _−M_ _[−][1]C_ � y y[′] �′ = � � y y[′] � _._ � ��A � b) A sajátértékek a det(A _−λI) polinom gyökei. Sor- és oszlopműveletekkel a determináns_ így egyszerűsíthető: _−λI_ _I_ det(A − _λI) =_ _−M_ _[−][1]K_ _−M_ _[−][1]C −_ _λI_ �� 0 _I_ = _−M_ _[−][1]K −_ _λM_ _[−][1]C −_ _λ[2]I_ _−M_ _[−][1]C −_ _λI_ �� 0 _I_ = (−1)[n] _M_ _[−][1]K + λM_ _[−][1]C + λ[2]I_ _M_ _[−][1]C + λI_ �� _I_ 0 = _M_ _[−][1]C + λI_ _M_ _[−][1]K + λM_ _[−][1]C + λ[2]I_ �� 1 = det(M _[−][1]K + λM_ _[−][1]C + λ[2]I) =_ det M [det(][λ][2][M][ +][ λC][ +][ K][)][.] Az első lépésben az i. oszlophoz hozzáadtuk az i + n. oszlop λ-szorosát (i = 1, . . ., n), a másodikban az utolsó n sort megszoroztuk −1-gyel, ezután az i. és i + n. oszlopot felcseréltük (i = 1, . . ., n), végül az első n sor szerint kifejtettük a determinánst, és felhasználtuk a determináns multiplikativitását. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">akkor teljesül, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">x</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">x</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ez azt jelenti, hogy</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (és</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> x</span></b></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">is)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátvektora</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátértékkel. Ha választunk egy bázist</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátvektoraiból, akkor</span></p> <p style="top:88.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">így képezhetünk</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sajátvektoraiból álló bázist is, amivel az általános megoldás a szoká-</span></p> <p style="top:102.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sos módon felírható. Legyenek</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">lineárisan független sajátvektorai</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . ,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> v</span></b><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a hozzájuk</span></p> <p style="top:116.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tartozó sajátértékek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ω</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , ω</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ekkor az általános megoldás</span></p> <p style="top:149.5pt;left:106.4pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:139.8pt;left:149.7pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></p> <p style="top:141.2pt;left:145.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">X</span></p> <p style="top:164.6pt;left:145.4pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">=1</span></p> <p style="top:149.5pt;left:159.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">ω</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">v</span></b><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:183.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Egy másik lehetőség: az elsőrendű esethez hasonlóan itt is gondolhatunk az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> speciális</span></p> <p style="top:198.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">esetre, ekkor tudjuk, hogy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> cos</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adja a megoldást. Hatványsorral ezek a függvények</span></p> <p style="top:212.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">is értelmezhetőek mátrix argumentumra (ehhez még az sem szükséges, hogy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pozitív</span></p> <p style="top:227.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">deﬁnit vagy szimmetrikus legyen!), az általános megoldás</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = cos(Ω</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+sin(Ω</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)Ω</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:233.1pt;left:530.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:241.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Persze ennek kiszámításához szintén</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ω</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátértékeit és sajátvektorait érdemes megkeresni.</span></p> <p style="top:260.8pt;left:56.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">12.</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∗</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Legyenek</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M, C, K n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">×</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-es mátrixok,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> invertálható, és tekintsük az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:275.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">másodrendű diﬀerenciálegyenlet-rendszert.</span></p> <p style="top:289.7pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) Írjuk át elsőrendű egyenletrendszerré az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">komponenseit tartalmazó (</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> elemű) vek-</span></p> <p style="top:304.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">torértékű függvényre nézve.</span></p> <p style="top:320.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) Mutassuk meg, hogy az így kapott elsőrendű állandó együtthatós diﬀerenciálegyenlet-</span></p> <p style="top:335.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">rendszer mátrixának sajátértékei a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λC</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyenlet gyökei. (Az ilyen</span></p> <p style="top:349.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">típusú egyenletek neve kvadratikus sajátérték-probléma.)</span></p> <p style="top:365.9pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) Tegyük fel, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M, C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mindegyike pozitív deﬁnit. Mutassuk meg, hogy ekkor</span></p> <p style="top:380.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">minden sajátérték valós része negatív. (Ebből következik, hogy minden megoldás</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-hoz</span></p> <p style="top:394.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tart amint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.)</span></p> <p style="top:414.3pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:428.7pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) Az elsőrendű egyenletrendszer blokk-mátrix alakban így írható fel:</span></p> <p style="top:447.9pt;left:126.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:456.5pt;left:134.1pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:471.0pt;left:132.7pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:447.9pt;left:142.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></p> <p style="top:463.8pt;left:154.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:447.9pt;left:167.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:456.5pt;left:191.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:456.5pt;left:245.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:471.0pt;left:172.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i></p> <p style="top:471.0pt;left:226.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i></p> <p style="top:447.9pt;left:269.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:481.6pt;left:167.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">\|</span></p> <p style="top:481.6pt;left:216.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">{z</span></p> <p style="top:481.6pt;left:270.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">}</span></p> <p style="top:493.3pt;left:217.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">A</span></i></p> <p style="top:447.9pt;left:277.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:456.5pt;left:284.2pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b></p> <p style="top:471.0pt;left:282.8pt;line-height:12.0pt"><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:447.9pt;left:292.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:463.8pt;left:300.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:513.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) A sajátértékek a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> polinom gyökei. Sor- és oszlopműveletekkel a determináns</span></p> <p style="top:527.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">így egyszerűsíthető:</span></p> <p style="top:560.6pt;left:128.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:544.3pt;left:206.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:553.3pt;left:220.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:553.3pt;left:295.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:567.8pt;left:209.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i></p> <p style="top:567.8pt;left:263.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:544.3pt;left:333.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:594.5pt;left:193.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:578.2pt;left:206.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:587.2pt;left:272.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:587.2pt;left:382.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:601.6pt;left:209.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λM</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:601.6pt;left:350.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i></p> <p style="top:578.2pt;left:420.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:628.4pt;left:193.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup></p> <p style="top:612.1pt;left:238.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:621.1pt;left:299.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:621.1pt;left:399.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:635.5pt;left:241.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λM</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:635.5pt;left:372.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λI</span></i></p> <p style="top:612.1pt;left:433.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:662.3pt;left:193.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:645.9pt;left:206.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:654.9pt;left:236.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:654.9pt;left:337.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:669.4pt;left:209.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λI</span></i></p> <p style="top:669.4pt;left:280.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λM</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i></p> <p style="top:645.9pt;left:401.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:693.8pt;left:193.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">K</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λM</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">I</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:685.7pt;left:382.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:702.0pt;left:369.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">det</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> M</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">det(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">M</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λC</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> K</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:723.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az első lépésben az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> oszlophoz hozzáadtuk az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> oszlop</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-szorosát (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">),</span></p> <p style="top:738.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a másodikban az utolsó</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sort megszoroztuk</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-gyel, ezután az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> oszlopot</span></p> <p style="top:752.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felcseréltük (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), végül az első</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sor szerint kifejtettük a determinánst, és</span></p> <p style="top:767.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">felhasználtuk a determináns multiplikativitását.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> </div>	page_318.png	akkor teljesül, ha Xx Ax, tehát —x?y — fily. Ez azt jelenti, hogy y (és x ís) ( sajátvoktora kl. Ha választunk egy bázist ) sajátvektoraiból, akkor yíz) - (G cosleuzjv, 4 D, sin(ezhv,). us legyen), az általános megoldás y(x) — cos(Ujyossin(20-ty. definít vagy s Persze ennek ki 12. s Legyenek M. C. K nc n-es mátrixok, M invertálható, és tekintsük az My"-Cy/4-Ky másodrendű differenciálegyenlet egyenletrendszerré az y. y/ komponenseit tartalmazó (2n el dszert. rendszer mátrixának sajátértékei a det(22M 4. 3C 4. ) — 0 egyenlet győkei. (Az ilye tart amint 2 — 06.) . Megoldás. a) Az elsőrendű egyenletrendszer blokk-mátrix alakban így írható fel 1) A sajátértékek a det( A— A/) polinom gyökei. Sor- és oszlapműveletekkel a determináns Így egyszerűsíthető: 1 1 ARME MCM ! 0. 1 [] MIGCGYAL MÓKGAMICA L s detM-AK 4 AMIC 4 37) — aogyy Jet0M 430 410. felhasználtuk a determináns multiplikatívitását.
tehát y 7→p(y) az új ismeretlen függvény. Ekkor y′′(x) = p′(y(x))y′(x) = p′(y(x))p(y(x)), így behelyettesítés után a pp′ = f(y, p) q elsőrendű diﬀerenciálegyenlethez jutunk. Ennek megoldása után az y′ = p(y) szétválasztható diﬀerenciálegyenletet kell megoldani. Speciális eset: ha f csak az első változótól függ, akkor egy −U primitív függvényt választva (U ′(y) = −f(y, y′)) az első diﬀerenciálegyenlet megoldása p(y) = 2(E −U(y)), ahol az E paraméter neve energia, U(y) pedig a potenciális energia. Oldjuk meg az alábbi másodrendű autonóm diﬀerenciálegyenleteket. a) y′′(1 + y2) = yy′2, b) y′′ = y′2 y y′2 y + y′ √ln y az y(0) = e, y′(0) = 2e kezdeti feltétel mellett, y c) y′′ = 1 y3, d) y′′ = −y. További gyakorló feladatok 8. Stabilis-e az y′ 1 y′ 1 = −y1 + y3 + y4 y′ 2 = −2y1 + y2 −2 y′ 2 = −2y1 + y2 −2y4 y′ 3 = −y2 + y3 + 2y4 y′ 3 = −y2 + y3 + 2y4 y′ 4 = −2y1 + y2 −y4 y′ 4 = −2y1 + y2 −y4 diﬀerenciálegyenlet-rendszer? 9. Határozzuk meg az alábbi diﬀerenciálegyenlet-rendszerek stacionárius pontjait. Milyen típusúak stabilitás tekintetében? a) y′ 2 1 = −2y1 + y3 y′ 2 = −2y1 + y2 b) y′ 1 y′ 1 1 = y2 y′ 1 = y2 1 + y2 −y1y2 −2 y′ 1 −2y2 + y1y2 + 2 = −y2 y′ 1 2 = −y2 y2 1 −2y2 + y1y2 + 4 c) y′ 1 y′ 1 = sin y1 + sin y2 y′ 2 = −sin y1 + sin y′ 2 = −sin y1 + sin y2 d) y′ 1 y′ 1 1 = (1 −y2 y2 2 1 −y2 y′ 1 −y2 1 = (1 −y2 2)y1 y′ 2)y2 1 −y2 2 = (1 −y2 y′ 1 2 = (1 −y2 y2 2 1 −y2 y2 2)y2 e) y′ 1 y′ 2 1 = (y1 −1)2 + y2 y′ 1 = (y1 −1)2 + y2 2 −2 y′ 2 −2 2 = (y1 + 1)2 + y2 1 2 y′ 2 2 = (y1 + 1)2 + y2 y2 2 −2	tehát y �→ _p(y) az új ismeretlen függvény. Ekkor y[′′](x) = p[′](y(x))y[′](x) = p[′](y(x))p(y(x)),_ így behelyettesítés után a _pp[′]_ = f (y, p) elsőrendű differenciálegyenlethez jutunk. Ennek megoldása után az y[′] = p(y) szétválasztható differenciálegyenletet kell megoldani. Speciális eset: ha f csak az első változótól függ, akkor egy −U primitív függvényt választva (U _[′](y) = −f_ (y, y[′])) az első differenciálegyenlet megoldása p(y) = �2(E − _U_ (y)), ahol az E paraméter neve energia, U (y) pedig a potenciális energia. Oldjuk meg az alábbi másodrendű autonóm differenciálegyenleteket. a) y[′′](1 + y[2]) = yy[′][2], _y[′]_ b) y[′′] = _[y]y[′][2]_ [+] _√ln y az y(0) = e, y[′](0) = 2e kezdeti feltétel mellett,_ c) y[′′] = 1 _y[3]_ [,] d) y[′′] = −y. ## További gyakorló feladatok 8. Stabilis-e az _y1[′]_ [=][ −][y][1] [+][ y][3] [+][ y][4] _y2[′]_ [=][ −][2][y][1] [+][ y][2] _[−]_ [2][y][4] _y3[′]_ [=][ −][y][2] [+][ y][3] [+ 2][y][4] _y4[′]_ [=][ −][2][y][1] [+][ y][2] _[−]_ _[y][4]_ differenciálegyenlet-rendszer? 9. Határozzuk meg az alábbi differenciálegyenlet-rendszerek stacionárius pontjait. Milyen típusúak stabilitás tekintetében? a) _y1[′]_ [=][ −][2][y][1] [+][ y]2[3] _y2[′]_ [=][ −][2][y][1] [+][ y][2] b) _y1[′]_ [=][ y]1[2] [+][ y][2] _[−]_ _[y][1][y][2]_ _[−]_ [2] _y2[′]_ [=][ −][y]1[2] _[−]_ [2][y][2] [+][ y][1][y][2] [+ 4] c) _y1[′]_ [= sin][ y][1] [+ sin][ y][2] _y2[′]_ [=][ −] [sin][ y][1] [+ sin][ y][2] d) _y1[′]_ [= (1][ −] _[y]1[2]_ _[−]_ _[y]2[2][)][y][1]_ _y2[′]_ [= (1][ −] _[y]1[2]_ _[−]_ _[y]2[2][)][y][2]_ e) _y1[′]_ [= (][y][1] _[−]_ [1)][2][ +][ y]2[2] _[−]_ [2] _y2[′]_ [= (][y][1] [+ 1)][2][ +][ y]2[2] _[−]_ [2] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> 7→</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az új ismeretlen függvény. Ekkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">így behelyettesítés után a</span></p> <p style="top:96.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">pp</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:118.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elsőrendű diﬀerenciálegyenlethez jutunk. Ennek megoldása után az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szétválaszt-</span></p> <p style="top:133.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ható diﬀerenciálegyenletet kell megoldani.</span></p> <p style="top:147.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Speciális eset: ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> csak az első változótól függ, akkor egy</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> primitív függvényt választva</span></p> <p style="top:163.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) az első diﬀerenciálegyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:153.2pt;left:409.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:163.6pt;left:419.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> E</span></i></p> <p style="top:179.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paraméter neve energia,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pedig a potenciális energia.</span></p> <p style="top:193.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Oldjuk meg az alábbi másodrendű autonóm diﬀerenciálegyenleteket.</span></p> <p style="top:208.0pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:232.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:241.0pt;left:129.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:224.7pt;left:165.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:232.2pt;left:156.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett,</span></p> <p style="top:256.5pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:255.0pt;left:128.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:263.8pt;left:125.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:274.1pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:306.3pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:330.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Stabilis-e az</span></p> <p style="top:352.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:359.1pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:370.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:376.5pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:387.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:393.9pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:405.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:411.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:427.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer?</span></p> <p style="top:444.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Határozzuk meg az alábbi diﬀerenciálegyenlet-rendszerek stacionárius pontjait.</span></p> <p style="top:444.2pt;left:503.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Milyen</span></p> <p style="top:458.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">típusúak stabilitás tekintetében?</span></p> <p style="top:473.1pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span></p> <p style="top:495.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:501.8pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:501.8pt;left:198.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:513.1pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:519.2pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:535.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span></p> <p style="top:558.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:564.3pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:564.3pt;left:158.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:575.6pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:581.8pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:581.8pt;left:168.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></sup></p> <p style="top:598.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span></p> <p style="top:620.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:626.9pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:638.1pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:644.3pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:660.7pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span></p> <p style="top:683.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:689.4pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:689.4pt;left:183.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:689.4pt;left:209.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:700.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:706.8pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:706.8pt;left:183.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:706.8pt;left:209.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:723.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e)</span></p> <p style="top:745.8pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:751.9pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:751.9pt;left:218.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:763.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:769.3pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:769.3pt;left:217.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_319.png	át y - píy) az új ismeretlen függyés így behelyettesítés után a Ekkor y"(z) — plylzhjye) — plylelhplyle). G.p) elsőrendű differenciálegyenlet! [ 2 futuak. Ennek megoldása után az 4 — ply) szétválaszt. ható dífferenciálegyenletet kell megoldati. Speciális eset: ha / csak az első változótól függ, akkor egy —U priaitív függyényt választva (U"() — —f(.17)) az első dítferenciálogyenlet megoldása píy) — 2(E — Úly)). ahol az E paraméter neve nergia, Ufy) pedig a potenciális energia. Oldjuk meg az alábbi másodrendű autonóm differenciálegyenleteket. HEZTSÉPŐ 1) 7 É sz yf0) — e. (0) — 2e kezdeti féltétel mellett FTI 94v-$ aY. "További gyakorló feladatok 8. Stabilis-e az HEZ ÉT 2n 492 — 291 2tn 42. 2m differenciálegyenle -rendszer? 9. Határozzuk meg az alábbi dillerenciálegyenlet-rendszerek stacionárius pontjait. . Milyen típusúak stabilitás tekintetében? a 2 412 2tw :1 úzAtn-un 2 W 2a nye t 4 9 snyi tn 4 ÁvdA vE 9 n1 2 úml
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 13. feladatsor: Stabilitásvizsgálat, speciális egyenlettípusok 1. Instabilis vagy stabilis az y′ 1 y′ 1 = y1 + 3y2 + 2y3 y′ 2 = −y2 −2y3 y′ 2 = −y2 −2y3 y′ 3 = 2y1 + 3y2 y′ 3 = 2y1 + 3y2 −y3 diﬀerenciálegyenlet-rendszer? Igaz-e, hogy aszimptotikusan stabilis? 2. Stabilis-e az y′ = Ay,   3 −9 9 −1 4 −5 −2 7 −8 A =   5 8 diﬀerenciálegyenlet-rendszer? 3. Hol vannak és milyen típusúak stabilitás tekintetében az y′ = y −y3 diﬀerenciálegyenlet stacionárius pontjai? 4. A csillapított síkinga mozgásegyenlete y′′ + 2αy′ + sin(y) = 0, ahol α > 0 jelent csillapítást, y a függőlegessel bezárt szög (y = 2kπ lefelé, y = (2k+1)π pedig felfelé). Mik az egyensúlyi pontok és melyek stabilisak? 5. Bernoulli-féle diﬀerenciálegyenletnek nevezzük az A1(x)y′ + A0(x)y = B(x)yα alakú egyenleteket, ahol α ∈R\{0, 1}, A0, A1 és B adott függvények. Az ilyen egyenleteket u(x) = y(x)1−α helyettesítéssel visszavezethetjük elsőrendű lineáris diﬀerenciálegyenletre. Oldjuk meg az alábbi Bernoulli-féle diﬀerenciálegyenleteket: a) 2y′ + y = (x −1)y3, b) 3(1 + x2)y′ + y = y4. 6. Euler-féle diﬀerenciálegyenletnek nevezzük az any(n) + an−1 1 xy(n−1) + · · · + a1 1 xn−1y′ + a0 1 xn = b(x) alakú lineáris diﬀerenciálegyenleteket, ahol a0, . . . , an valós számok, an ̸= 0 és b(x) adott függvény. A homogén egyenletet y(x) = u(ln \|x\|) helyettesítéssel visszavezethetjük állandó együtthatós homogén lineáris diﬀerenciálegyenletre. Oldjuk meg az alábbi Euler-féle diﬀerenciálegyenleteket: j a) y′′ + 5 g 5 xy′ + 4 x2 6 4 x2y = 0, 12 b) y(4) + 6 x 6 xy′′′ + 12 x2 6 12 x2 y′′ + 6 x3 6 x3y′ −36 x4 36 x4 y = 0, c) y′′ −6 x2 6 x2y = 5x. x 7. Az y′′ = f(y, y′) alakú egyenleteket másodrendű autonóm diﬀerenciálegyenletnek nevezzük. Az ilyen egyenletek megoldásánál érdemes y′(x) = p(y(x)) helyettesítést alkalmazni, ahol	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 13. feladatsor: Stabilitásvizsgálat, speciális egyenlettípusok 1. Instabilis vagy stabilis az _y1[′]_ [=][ y][1] [+ 3][y][2] [+ 2][y][3] _y2[′]_ [=][ −][y][2] _[−]_ [2][y][3] _y3[′]_ [= 2][y][1] [+ 3][y][2] _[−]_ _[y][3]_ differenciálegyenlet-rendszer? Igaz-e, hogy aszimptotikusan stabilis? 2. Stabilis-e az y[′] = Ay, _A =_  3 _−9_ 9  −1 4 _−5_ _−2_ 7 _−8_ differenciálegyenlet-rendszer? 3. Hol vannak és milyen típusúak stabilitás tekintetében az y[′] = y − _y[3]_ differenciálegyenlet stacionárius pontjai? 4. A csillapított síkinga mozgásegyenlete y[′′] + 2αy[′] + sin(y) = 0, ahol α > 0 jelent csillapítást, _y a függőlegessel bezárt szög (y = 2kπ lefelé, y = (2k_ +1)π pedig felfelé). Mik az egyensúlyi pontok és melyek stabilisak? 5. Bernoulli-féle differenciálegyenletnek nevezzük az _A1(x)y[′]_ + A0(x)y = B(x)y[α] alakú egyenleteket, ahol α ∈ R\{0, 1}, A0, A1 és B adott függvények. Az ilyen egyenleteket _u(x) = y(x)[1][−][α]_ helyettesítéssel visszavezethetjük elsőrendű lineáris differenciálegyenletre. Oldjuk meg az alábbi Bernoulli-féle differenciálegyenleteket: a) 2y[′] + y = (x − 1)y[3], b) 3(1 + x[2])y[′] + y = y[4]. 6. Euler-féle differenciálegyenletnek nevezzük az 1 1 1 _any[(][n][)]_ + an−1 _x_ _[y][(][n][−][1)][ +][ · · ·][ +][ a][1]_ _x[n][−][1]_ _[y][′][ +][ a][0]_ _x[n][ =][ b][(][x][)]_ alakú lineáris differenciálegyenleteket, ahol a0, . . ., an valós számok, an ̸= 0 és b(x) adott függvény. A homogén egyenletet y(x) = u(ln \|x\|) helyettesítéssel visszavezethetjük állandó együtthatós homogén lineáris differenciálegyenletre. Oldjuk meg az alábbi Euler-féle differenciálegyenleteket: a) y[′′] + [5] _x_ _[y][′][ + 4]x[2]_ _[y][ = 0,]_ b) y[(4)] + [6] _x_ _[y][′′′][ + 12]x[2][ y][′′][ + 6]x[3]_ _[y][′][ −]_ [36]x[4][ y][ = 0,] c) y[′′] _−_ [6] _x[2]_ _[y][ = 5][x][.]_ 7. Az y[′′] = f (y, y[′]) alakú egyenleteket másodrendű autonóm differenciálegyenletnek nevezzük. Az ilyen egyenletek megoldásánál érdemes y[′](x) = p(y(x)) helyettesítést alkalmazni, ahol -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">13. feladatsor: Stabilitásvizsgálat, speciális</span></b></p> <p style="top:106.3pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">egyenlettípusok</span></b></p> <p style="top:145.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Instabilis vagy stabilis az</span></p> <p style="top:172.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:178.3pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:189.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:195.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:207.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:213.2pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:233.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer? Igaz-e, hogy aszimptotikusan stabilis?</span></p> <p style="top:249.9pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Stabilis-e az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:289.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:267.9pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:285.4pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:275.3pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:275.3pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:275.3pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:289.7pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:289.7pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:289.7pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:304.2pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:304.2pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> <p style="top:304.2pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> <p style="top:267.9pt;left:203.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:285.4pt;left:203.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:329.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer?</span></p> <p style="top:346.1pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Hol vannak és milyen típusúak stabilitás tekintetében az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet</span></p> <p style="top:360.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">stacionárius pontjai?</span></p> <p style="top:377.0pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. A csillapított síkinga mozgásegyenlete</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelent csillapítást,</span></p> <p style="top:391.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a függőlegessel bezárt szög (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">kπ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> lefelé,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pedig felfelé). Mik az egyensúlyi</span></p> <p style="top:405.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pontok és melyek stabilisak?</span></p> <p style="top:422.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Bernoulli-féle diﬀerenciálegyenletnek nevezzük az</span></p> <p style="top:448.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></sup></p> <p style="top:475.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú egyenleteket, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\{</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">}</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adott függvények. Az ilyen egyenleteket</span></p> <p style="top:489.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyettesítéssel visszavezethetjük elsőrendű lineáris diﬀerenciálegyenletre.</span></p> <p style="top:504.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Oldjuk meg az alábbi Bernoulli-féle diﬀerenciálegyenleteket:</span></p> <p style="top:518.5pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:534.9pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:551.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6. Euler-féle diﬀerenciálegyenletnek nevezzük az</span></p> <p style="top:582.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:574.2pt;left:175.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:590.4pt;left:174.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> · · ·</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:574.2pt;left:275.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:590.4pt;left:266.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span></sup></p> <p style="top:574.2pt;left:329.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:590.4pt;left:326.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> b</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup></p> <p style="top:612.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú lineáris diﬀerenciálegyenleteket, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, . . . , a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> valós számok,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> a</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">n</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ̸</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> b</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adott</span></p> <p style="top:626.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">függvény. A homogén egyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> \|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítéssel visszavezethetjük állandó</span></p> <p style="top:641.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">együtthatós homogén lineáris diﬀerenciálegyenletre.</span></p> <p style="top:655.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Oldjuk meg az alábbi Euler-féle diﬀerenciálegyenleteket:</span></p> <p style="top:674.6pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></sup></p> <p style="top:682.8pt;left:124.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 4</span></sup></p> <p style="top:682.8pt;left:156.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:701.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(4)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></sup></p> <p style="top:709.6pt;left:130.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 12</span></sup></p> <p style="top:709.6pt;left:167.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 6</span></sup></p> <p style="top:709.6pt;left:207.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">36</span></sup></p> <p style="top:709.6pt;left:244.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:728.2pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></sup></p> <p style="top:736.5pt;left:124.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 5</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></sup></p> <p style="top:748.4pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakú egyenleteket másodrendű autonóm diﬀerenciálegyenletnek nevezzük.</span></p> <p style="top:762.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az ilyen egyenletek megoldásánál érdemes</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítést alkalmazni, ahol</span></p> </div>	page_320.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 13. feladatsor: Stabilitásvizsgálat, speciális egyenlettípusok 1. Tnstabilis vagy stabilis az DEPEZTEZTI W—W— 2 W 2n 43 — 5 dilferenciálegyet 2. Stabilis-e az y/ dszer? Igaz-e, hogy aszímptotikusan stabilis? A. k7 EEB] díferenciálegyenlet-rendszer? 3. Hol vannak és milyen típusúak stabilitás tekinte y— 47 dílfei stacionárius pontjai? 4. A csillapított sí w a függölegessel bozárt szóg (y pontok és melyek stabilisak? mazy álogyenlet ásegyenlete 4" 4-2ay! 4-sin(y) — 0, ahol a — 0 jelent csillapítást 2hr lefe 2k- 1) podig felfelé). Mik az egyensúlyi AV 4 Adlajy a B" alakú egyenleteket, ahol a € R 10.1), Av. A és B adott függvények. Az ilyen egyenleteket a(z) — v(z)172 helyettesítéssel visszavezethetjük elsőrendű lincáris differenciálegyenletre Oldjuk meg az alábbi Bernoulli-féle dífforenciálegyenleteket: a) 2/ 4 —(r 1 1) 31 44 é9 191 1 a 4 apazy 44 a t a— — Hr) fűggvény. A homogén egyenletet y(z) — u(ln [z]) helyettesítéssel visszavezethetjűk állandó együtthatós homogén lincáris differenciálegyenletre. Oldjuk meg az alábbi Eul -iálegyenleteket: 56. 4 NZEE , 64 , 12 , 5. 36 F2v : 1) 6 9Y-2áv 7. Az / — f(v.v) alakú egye teket másodrendű autonóm differenciálegyenletnek nevezzüűk. ést alkalmazni, ahol
b) A stacionárius pontok f(y1, y2) = (y2 1 y2 1 + y2 −y1y2 −2, −y2 1 há A stacionárius pontok f(y1, y2) = (y2 1 + y2 −y1y2 −2, −y2 1 −2y2 + y1y2 + 4) = (0, 0) megoldásai. A két egyenlet összege 2 −y2 = 0, tehát y2 = 2 és y1 = 0 vagy y1 = 2. A D(y1,y2)f lineáris leképezés mátrixa D(y1,y2) = " # , 2y1 −y2 1 −y1 −2y1 + y2 −2 + y2 amibe a stacionárius pontokat helyettesítve a D(0,2) = # −2 1 2 −2 és D(2,2) = # " 2 −1 −2 0 mátrixok adódnak. Ezek karakterisztikus polinomjai λ2 + 4λ + 2 és λ2 −2λ −2, tehát az első aszimptotikusan stabilis, a második instabilis stacionárius pont. c) 0 = sin y1 + sin y2 = −sin y1 + sin y2 pontosan akkor teljesül, ha sin y1 = sin y2 = 0, tehát ha y1 és y2 is a π egész számú többszöröse. A jobboldal deriváltmátrixa mátrixok adódnak. Ezek karakterisztikus polinomjai λ2 + 4λ + 2 és λ2 −2λ −2, tehát az első aszimptotikusan stabilis, a második instabilis stacionárius pont. c) 0 = sin y1 + sin y2 = −sin y1 + sin y2 pontosan akkor teljesül, ha sin y1 = sin y2 = 0, D(y1,y2) = " # , cos y1 cos y2 −cos y1 cos y2 aminek értéke az egyes a stacionárius pontokban −1 −1 " # 1 −1 −1 −1 " # −1 −1 1 −1 √ √ 2, ± aszerint, hogy y1 és y2 páros vagy páratlan többszöröse π-nek. A sajátértékpárok rendre 1±i, ± 2 illetve −1±i. Tehát az y1 = (2k+1)π, y2 = (2l+1)π alakú stacionárius √ √ 2, ± 1±i, ± 2 illetve −1±i. Tehát az y1 = (2k+1)π, y2 = (2l+1)π alakú stacionárius pontok aszimptotikusan stabilisak, a többi instabilis. d) A két egyenlet jobb oldala akkor 0, ha y2 1 + y2 2 = 1 vagy y1 = y2 = 0. A deriváltmátrix √ 2, ± y2 2 1 + y2 y2 2 = 1 vagy y1 = y2 = 0. A deriváltmátrix " 1 −3y2 1 y2 2 1 −y2 D(y1,y2) = −3y2 1 −y2 2 −2y1y2 −2y1y2 1 −y2 1 −3 y2 2 1 −3y2 aminek a karakterisztikus polinomja λ2 + 2(2y2 1 y2 2 1 + 2y2 y2 1 2 −1)λ + (y2 y2 2 1 + y2 y2 1 2 −1)(3y2 y2 2 1 + 3y2 y2 2 −1), így a sajátértékek 1 −y2 1 y2 2 1 −y2 y2 1 2 és 1 −3y2 y2 2 1 −3y2 így a sajátértékek 1 −y2 1 −y2 1 −3y2 2 és 1 −3y2 2. Az origóban mindkettő 1, tehát ez instabilis, az egységkörön pedig 0 és −2, tehát a stabilitás ennek alapján nem dönthető el (egyébként stabilis, de nem aszimptotikusan stabilis). e) A stacionárius pontok azok az (y1, y2) párok, amelyekre az egyenletek jobb oldala 0, p (y1, y2) p , y tehát (y1 −1)2 = (y1 + 1)2, amiből y1 = 0 és így 0 = 1 + y2 2 tehát (y1 −1)2 = (y1 + 1)2, amiből y1 = 0 és így 0 = 1 + y2 2 −2, tehát y2 = ±1. A jobb oldal deriváltmátrixa D(y1,y2) = " # , 2(y1 −1) 2y2 2(y1 + 1) 2y2 q ennek sajátértékei y1 +y2 −1± √ (y1 + y2 −1)2 + 8y2. Az (y1, y2) = (0, 1) pontban ezek ±2 2, tehát ez a pont instabilis, az (y1, y2) = (0, −1) pontban pedig −2 ± 2i, tehát ez ennek sajátértékei y1 +y2 −1± √ ±2 2, tehát ez a pont instabilis, az (y1, y2) = (0, −1) pontban pedig −2 ± 2i, tehát ez a pont aszimptotikusan stabilis.	b) A stacionárius pontok f (y1, y2) = (y1[2] [+][ y][2] _[−]_ _[y][1][y][2]_ _[−]_ [2][,][ −][y]1[2] _[−]_ [2][y][2] [+][ y][1][y][2] [+ 4) = (0][,][ 0)] megoldásai. A két egyenlet összege 2 − _y2 = 0, tehát y2 = 2 és y1 = 0 vagy y1 = 2. A_ _D(y1,y2)f lineáris leképezés mátrixa_ _D(y1,y2) =_ � 2y1 − _y2_ 1 − _y1_ _−2y1 + y2_ _−2 + y2_ � _,_ amibe a stacionárius pontokat helyettesítve a � 2 _−1�_ _−2_ 0 _D(0,2) =_ �−2 1 � és D(2,2) = 2 _−2_ mátrixok adódnak. Ezek karakterisztikus polinomjai λ[2] + 4λ + 2 és λ[2] _−_ 2λ − 2, tehát az első aszimptotikusan stabilis, a második instabilis stacionárius pont. c) 0 = sin y1 + sin y2 = − sin y1 + sin y2 pontosan akkor teljesül, ha sin y1 = sin y2 = 0, tehát ha y1 és y2 is a π egész számú többszöröse. A jobboldal deriváltmátrixa � _,_ _D(y1,y2) =_ � cos y1 cos y2 _−_ cos y1 cos y2 aminek értéke az egyes a stacionárius pontokban � 1 1� �−1 1� � 1 _−1�_ �−1 _−1�_ _−1_ 1 1 1 _−1_ _−1_ 1 _−1_ aszerint, hogy√ _√ y1 és y2 páros vagy páratlan többszöröse π-nek. A sajátértékpárok rendre_ 1±i, ± 2, ± 2 illetve −1±i. Tehát az y1 = (2k+1)π, y2 = (2l+1)π alakú stacionárius pontok aszimptotikusan stabilisak, a többi instabilis. d) A két egyenlet jobb oldala akkor 0, ha y1[2] [+][ y]2[2] [= 1 vagy][ y][1] [=][ y][2] [= 0. A deriváltmátrix] � _,_ _D(y1,y2) =_ �1 − 3y1[2] _[−]_ _[y]2[2]_ _−2y1y2_ _−2y1y2_ 1 − _y1[2]_ _[−]_ [3][y]2[2] aminek a karakterisztikus polinomja _λ[2]_ + 2(2y1[2] [+ 2][y]2[2] _[−]_ [1)][λ][ + (][y]1[2] [+][ y]2[2] _[−]_ [1)(3][y]1[2] [+ 3][y]2[2] _[−]_ [1)][,] így a sajátértékek 1 − _y1[2]_ _[−]_ _[y]2[2]_ [és 1][ −] [3][y]1[2] _[−]_ [3][y]2[2][. Az origóban mindkettő 1, tehát ez] instabilis, az egységkörön pedig 0 és −2, tehát a stabilitás ennek alapján nem dönthető el (egyébként stabilis, de nem aszimptotikusan stabilis). e) A stacionárius pontok azok az (y1, y2) párok, amelyekre az egyenletek jobb oldala 0, tehát (y1 − 1)[2] = (y1 + 1)[2], amiből y1 = 0 és így 0 = 1 + y2[2] _[−]_ [2, tehát][ y][2] [=][ ±][1. A jobb] oldal deriváltmátrixa � _,_ _D(y1,y2) =_ �2(y1 − 1) 2y2 2(y1 + 1) 2y2 � ennek sajátértékei y1 + _y2 −_ 1 _±_ (y1 + y2 − 1)[2] + 8y2. Az (y1, y2) = (0, 1) pontban ezek _√_ _±2_ 2, tehát ez a pont instabilis, az (y1, y2) = (0, −1) pontban pedig −2 ± 2i, tehát ez a pont aszimptotikusan stabilis. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) A stacionárius pontok</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> f</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:65.2pt;left:285.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:65.2pt;left:394.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4) = (0</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span></sup></p> <p style="top:73.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldásai. A két egyenlet összege</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vagy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A</span></p> <p style="top:88.0pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">f</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> lineáris leképezés mátrixa</span></p> <p style="top:122.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:106.8pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:115.4pt;left:188.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:115.4pt;left:249.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:129.8pt;left:184.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:129.8pt;left:244.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:106.8pt;left:284.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:122.7pt;left:292.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:155.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">amibe a stacionárius pontokat helyettesítve a</span></p> <p style="top:189.1pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(0</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:173.1pt;left:170.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:181.7pt;left:176.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:181.7pt;left:205.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:196.2pt;left:180.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:196.2pt;left:201.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:173.1pt;left:216.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:189.1pt;left:228.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:173.1pt;left:285.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:181.7pt;left:295.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:181.7pt;left:316.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:196.2pt;left:291.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:196.2pt;left:320.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:173.1pt;left:331.3pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:223.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mátrixok adódnak. Ezek karakterisztikus polinomjai</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></p> <p style="top:237.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az első aszimptotikusan stabilis, a második instabilis stacionárius pont.</span></p> <p style="top:254.3pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 = sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontosan akkor teljesül, ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = sin</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:268.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> is a</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egész számú többszöröse. A jobboldal deriváltmátrixa</span></p> <p style="top:302.0pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:286.0pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:294.7pt;left:190.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:294.7pt;left:233.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:309.1pt;left:184.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:309.1pt;left:233.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:286.0pt;left:261.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:302.0pt;left:269.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:335.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">aminek értéke az egyes a stacionárius pontokban</span></p> <p style="top:352.4pt;left:126.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:361.0pt;left:137.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:361.0pt;left:157.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:132.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:157.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:352.4pt;left:163.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:352.4pt;left:183.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:361.0pt;left:189.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:361.0pt;left:214.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:193.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:214.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:352.4pt;left:220.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:352.4pt;left:239.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:361.0pt;left:250.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:361.0pt;left:270.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:245.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:270.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:352.4pt;left:285.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:352.4pt;left:305.1pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:361.0pt;left:310.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:361.0pt;left:336.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:315.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:375.5pt;left:336.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:352.4pt;left:351.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:401.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">aszerint, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> páros vagy páratlan többszöröse</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-nek. A sajátértékpárok rendre</span></p> <p style="top:415.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:406.2pt;left:133.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:415.9pt;left:143.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:406.2pt;left:164.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:415.9pt;left:174.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> illetve</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Tehát az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">l</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakú stacionárius</span></p> <p style="top:430.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pontok aszimptotikusan stabilisak, a többi instabilis.</span></p> <p style="top:446.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) A két egyenlet jobb oldala akkor</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:453.0pt;left:303.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:453.0pt;left:328.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> vagy</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A deriváltmátrix</span></sup></p> <p style="top:480.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:464.7pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:473.4pt;left:184.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:479.5pt;left:216.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:479.5pt;left:241.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:473.4pt;left:270.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:487.8pt;left:197.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:487.8pt;left:256.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:493.9pt;left:283.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:493.9pt;left:314.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:464.7pt;left:319.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:480.7pt;left:327.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:513.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">aminek a karakterisztikus polinomja</span></p> <p style="top:540.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2(2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:546.4pt;left:174.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:546.4pt;left:206.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:546.4pt;left:267.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:546.4pt;left:293.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)(3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:546.4pt;left:339.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:546.4pt;left:370.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:566.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">így a sajátértékek</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:572.8pt;left:222.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:572.8pt;left:249.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:572.8pt;left:307.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:572.8pt;left:340.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az origóban mindkettő</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát ez</span></sup></p> <p style="top:581.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">instabilis, az egységkörön pedig</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát a stabilitás ennek alapján nem dönthető</span></p> <p style="top:595.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">el (egyébként stabilis, de nem aszimptotikusan stabilis).</span></p> <p style="top:612.0pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e) A stacionárius pontok azok az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> párok, amelyekre az egyenletek jobb oldala</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:626.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és így</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 = 1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:632.5pt;left:391.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A jobb</span></sup></p> <p style="top:640.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">oldal deriváltmátrixa</span></p> <p style="top:671.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:655.8pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:664.4pt;left:184.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></p> <p style="top:664.4pt;left:240.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:678.9pt;left:184.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span></p> <p style="top:678.9pt;left:240.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:655.8pt;left:256.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:671.7pt;left:264.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:709.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek sajátértékei</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i></p> <p style="top:699.0pt;left:255.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:709.4pt;left:265.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 8</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban ezek</span></p> <p style="top:726.7pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:716.9pt;left:112.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:726.7pt;left:122.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát ez a pont instabilis, az</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pontban pedig</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát ez</span></p> <p style="top:741.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a pont aszimptotikusan stabilis.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> </div>	page_321.png	1) A stacionárius pontok F( 2) ( 42 — nya — 2,—yl — 22 4 Inya 4 0. tehát ya — 2 és megoldásai. A két egyenlet összege 2 — 4 Diy s lineáris leképezés mátrixa Díya) — amibe a stacionárius pontokat helyettesítve a. mátrixok adódnak. Ezek karakterisztikus polinomjai ? 443 4.2 és 32 — 23— 2, tehát e) 0 — sinyi - sinye — — siny 4-sin ye pontosan akkor teljesül, ha sin m n D— [ eosm . cos; v,v] eosy cosya aminek értéke az egyes a stacionárius pontokban 1) [] [1 -] [- - 1aj [1 j [ -] [1 aszerint, hogy u és y, páros vagy páratlan többszöröse 7-nek. A sajátérti 124, 2423. 2/ illetve —12ti. Tehát az y — (2k- 17. 47 cpárok rendre. (21--1)2 alakú stacionárius d) A két egyenlet jobb oldala akkor 0, ha y7 4- 47 — 1 vagy 1 0. A deriváltmátrix 131 —42 22 aminek a karakterisztikus polinomja. 8 4202 4 22 — 19X 4 v 448 — (GE 434—1), így a sajátértékek 1— 48 — 43 és 1— 348 — 348. Az origóban mindkettő 1, tehát ez el (egyébkés 91 aszimptotikusan stabilis). tehát (yi — 192 — (y, 4192. amiből , — 0 és így 0 — 14-4§—2, tehát 4 — 21. A jobb 2, tehát a stabilitás em stabilis, de n ennek sajátértékei y, 4-99—142 // 4 92 — 192 3 892. Az (, 1) — (0.1) pontban ezek 4247. tehát ez a pont instabilis, az (y..92) — (0, —1) pontban pedig —24-21, tehát ez
Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz 13. feladatsor: Stabilitásvizsgálat, speciális egyenlettípusok (megoldás) 1. Instabilis vagy stabilis az y′ 1 y′ 1 = y1 + 3y2 + 2y3 y′ 2 = −y2 −2y3 y′ 2 = −y2 −2y3 y′ 3 = 2y1 + 3y2 y′ 3 = 2y1 + 3y2 −y3 diﬀerenciálegyenlet-rendszer? Igaz-e, hogy aszimptotikusan stabilis? Megoldás. y = (y1, y2, y3) jelöléssel az egyenlet y′ = Ay alakú, ahol  −1 −2 3 −1 A =  det(A−λI) = −λ3−λ2−λ−1 = −(λ+1)(λ2+1) gyökei −1 és ±i, tehát az egyenletrendszer stabilis, de nem aszimptotikusan stabilis. 2. Stabilis-e az y′ = Ay,   3 −9 9 −1 4 −5 −2 7 −8 A =   5 8 diﬀerenciálegyenlet-rendszer? Megoldás. det(A −λI) = −λ3 −λ2 = −λ2(λ + 1), tehát λ = 0 kétszeres, λ = −1 egyszeres sajátérték. Az utóbbival nem kell foglalkozni, az előbbihez tartozó sajátvektorok   x1 x2 x3 = 0 alapján (3, 2, 1) többszörösei, vagyis csak egydimenziós alteret alkotnak. Eszerint a 0 sajátértékhez egy 2 × 2 Jordan-blokk tartozik, vagyis az egyenletrendszer instabilis. 3. Hol vannak és milyen típusúak stabilitás tekintetében az y′ = y −y3 diﬀerenciálegyenlet stacionárius pontjai? Megoldás. y stacionárius pont, ha 0 = f(y) = y −y3 = y(1 −y2), vagyis y ∈{−1, 0, 1}. Mivel f ′(y) = 1 −3y2, és így f ′(0) = 1, f ′(±1) = −2, a 0 instabilis, ±1 aszimptotikusan stabilis. 4. A csillapított síkinga mozgásegyenlete y′′ + 2αy′ + sin(y) = 0, ahol α > 0 jelent csillapítást, y a függőlegessel bezárt szög (y = 2kπ lefelé, y = (2k+1)π pedig felfelé). Mik az egyensúlyi pontok és melyek stabilisak? Megoldás. y1 = y, y2 = y′ bevezetésével az y′ 1 y′ 1 = y2 y′ 2 = − y′ 2 = −sin(y1) −2αy2	# Matematika A3 gyakorlat Energetika és Mechatronika BSc szakok, 2016/17 ősz # 13. feladatsor: Stabilitásvizsgálat, speciális egyenlettípusok (megoldás) 1. Instabilis vagy stabilis az _y1[′]_ [=][ y][1] [+ 3][y][2] [+ 2][y][3] _y2[′]_ [=][ −][y][2] _[−]_ [2][y][3] _y3[′]_ [= 2][y][1] [+ 3][y][2] _[−]_ _[y][3]_ differenciálegyenlet-rendszer? Igaz-e, hogy aszimptotikusan stabilis? _Megoldás. y = (y1, y2, y3) jelöléssel az egyenlet y[′]_ = Ay alakú, ahol _A =_ 1 3 2  0 _−1_ _−2_ 2 3 _−1_ det(A−λI) = −λ[3] _−λ[2]_ _−λ−1 = −(λ+1)(λ[2]_ +1) gyökei −1 és ±i, tehát az egyenletrendszer stabilis, de nem aszimptotikusan stabilis. 2. Stabilis-e az y[′] = Ay, _A =_  3 _−9_ 9  −1 4 _−5_ _−2_ 7 _−8_ differenciálegyenlet-rendszer? _Megoldás. det(A −_ _λI) = −λ[3]_ _−_ _λ[2]_ = −λ[2](λ + 1), tehát λ = 0 kétszeres, λ = −1 egyszeres sajátérték. Az utóbbival nem kell foglalkozni, az előbbihez tartozó sajátvektorok _A_ x1 x2 _x3_   = 0 alapján (3, 2, 1) többszörösei, vagyis csak egydimenziós alteret alkotnak. Eszerint a 0 sajátértékhez egy 2 × 2 Jordan-blokk tartozik, vagyis az egyenletrendszer instabilis. 3. Hol vannak és milyen típusúak stabilitás tekintetében az y[′] = y − _y[3]_ differenciálegyenlet stacionárius pontjai? _Megoldás. y stacionárius pont, ha 0 = f_ (y) = y − _y[3]_ = y(1 − _y[2]), vagyis y ∈{−1, 0, 1}._ Mivel f _[′](y) = 1 −_ 3y[2], és így f _[′](0) = 1, f_ _[′](±1) = −2, a 0 instabilis, ±1 aszimptotikusan_ stabilis. 4. A csillapított síkinga mozgásegyenlete y[′′] + 2αy[′] + sin(y) = 0, ahol α > 0 jelent csillapítást, _y a függőlegessel bezárt szög (y = 2kπ lefelé, y = (2k_ +1)π pedig felfelé). Mik az egyensúlyi pontok és melyek stabilisak? _Megoldás. y1 = y, y2 = y[′]_ bevezetésével az _y1[′]_ [=][ y][2] _y2[′]_ [=][ −] [sin(][y][1][)][ −] [2][αy][2] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:54.9pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">Matematika A3 gyakorlat</span></b></p> <p style="top:73.5pt;left:56.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Energetika és Mechatronika BSc szakok, 2016/17 ősz</span></p> <p style="top:90.2pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">13. feladatsor: Stabilitásvizsgálat, speciális</span></b></p> <p style="top:107.5pt;left:56.7pt;line-height:17.2pt"><b><span style="font-family:LMRoman12,serif;font-size:17.2pt;color:#000000">egyenlettípusok (megoldás)</span></b></p> <p style="top:147.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1. Instabilis vagy stabilis az</span></p> <p style="top:174.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:180.1pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:191.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:197.6pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:208.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:215.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:235.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer? Igaz-e, hogy aszimptotikusan stabilis?</span></p> <p style="top:254.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelöléssel az egyenlet</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakú, ahol</span></p> <p style="top:295.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:273.1pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:290.6pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:280.4pt;left:137.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:280.4pt;left:158.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:280.4pt;left:183.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:294.9pt;left:137.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:294.9pt;left:153.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:294.9pt;left:178.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:309.3pt;left:137.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:309.3pt;left:158.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:309.3pt;left:178.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:273.1pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:290.6pt;left:193.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:336.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> gyökei</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát az egyenletrendszer</span></p> <p style="top:350.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">stabilis, de nem aszimptotikusan stabilis.</span></p> <p style="top:369.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2. Stabilis-e az</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:409.6pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:387.7pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:405.2pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:395.0pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span></p> <p style="top:395.0pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:395.0pt;left:192.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9</span></p> <p style="top:409.5pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:409.5pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> <p style="top:409.5pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> <p style="top:423.9pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:423.9pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7</span></p> <p style="top:423.9pt;left:187.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8</span></p> <p style="top:387.7pt;left:203.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:405.2pt;left:203.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:449.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer?</span></p> <p style="top:468.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kétszeres,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> egyszeres</span></p> <p style="top:483.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sajátérték. Az utóbbival nem kell foglalkozni, az előbbihez tartozó sajátvektorok</span></p> <p style="top:523.0pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i></p> <p style="top:501.0pt;left:117.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:518.6pt;left:117.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:508.4pt;left:123.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:522.9pt;left:123.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:537.3pt;left:123.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:501.0pt;left:135.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:518.6pt;left:135.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></p> <p style="top:563.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alapján</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> többszörösei, vagyis csak egydimenziós alteret alkotnak. Eszerint a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sa-</span></p> <p style="top:577.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">játértékhez egy</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Jordan-blokk tartozik, vagyis az egyenletrendszer instabilis.</span></p> <p style="top:597.2pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3. Hol vannak és milyen típusúak stabilitás tekintetében az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet</span></p> <p style="top:611.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">stacionárius pontjai?</span></p> <p style="top:631.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> stacionárius pont, ha</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagyis</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈{−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">}</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:645.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Mivel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, és így</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> instabilis,</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> aszimptotikusan</span></p> <p style="top:660.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">stabilis.</span></p> <p style="top:679.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4. A csillapított síkinga mozgásegyenlete</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelent csillapítást,</span></p> <p style="top:693.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a függőlegessel bezárt szög (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">kπ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> lefelé,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pedig felfelé). Mik az egyensúlyi</span></p> <p style="top:708.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">pontok és melyek stabilisak?</span></p> <p style="top:727.6pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">bevezetésével az</span></p> <p style="top:753.9pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:760.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:771.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:777.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">αy</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> </div>	page_322.png	Matematika A3 gyakorlat Energetika és Mechatzonika BSc szakok, 2016/17 ősz 13. feladatsor: Stabilitásvizsgálat, speciális egyenlettípusok (megoldás) 1. Tnstabilis vagy stabilis az HETEETETT] W—W— 2 2n t 3s dilferenciálegyet dszer? Igaz-e, hogy aszímptotikusan stabilis? . Megoldás. y — (y1.y2,11) jelöléssel az egyenlet Y" Ay alakú, ahol 15 2 oó 2] e 2] 38811 — O(X 1) gyökei —1 és 2-i, tehát az egyenletrendszer 2. Stabilis-e az Y — Ay, k7 EEB] dilferenciálegyet dszer? " Megoldás. det( A — AY a —X — 98 — —32X 4-1), tehát ) — 0 kétszeres, A — —1 egyszeres sajátérték. Az utóbbival nem kell foglalkozt az előbbihez tartozó sajátvektorok /lLu z0 alapján (3.2.1) többszörősei, vagyis csak egydimenziós alteret alkotnak. Eszerint a 0 sa- játértékhez egy 2 x 2 Jordan-blokk tartozik, vagyis az egyenletrendszer instabilis. 3. Hol vannak és milyen típusúak stabilitás tekintetél stacionárius pontjai? m az Y — y—17 differenciálegyenlet .Megoldás. y stacionárius pont, ha 0 Mivel ffly) — 1— 34. F stabilis. ) 2 — — V(— 1). vagyis y € (-1.0.1). 1. fűlae1) — 22, a 0 instabilis, 221 aszímptotikusan 4. A csillapított síkinga mozgásegyenlete y" 4-2ay/4-sin(y) w a függölegessel bezárt szög ( — 2lr lefe pontok és melyek stabilisak? ahol a —. 0 jelent csillapítást. 42 (2k--1)z pedig felfelé). Mik az egyensúlyi . Megoldás. v. w bevezetésével az m úzn 1— —sin() — 2a9
egyenletrendszert kapjuk, a jobb oldal akkor 0, ha y2 = 0 és −sin(y1) = 0, azaz y1 = kπ valamilyen k ∈Z-re. A deriváltmátrix " # , 0 1 −cos(y1) −2α √ q ennek sajátértékei −α ± α2 −cos(y1). Ha y1 = 2kπ, akkor ez −α ± √ α2 + 1, vagyis az egyensúlyi helyzet bármilyen α2 −1, tehát α > 0 esetén aszimptotikusan stabilis, α = 0 esetén a stabilitás nem dönthető el ezek alapján (egyébként stabilis, de nem aszimptotikusan stabilis), α < 0 esetén instabilis. Ha viszont y1 = (2k + 1)π, akkor a sajátértékek −α ± ennek sajátértékei −α ± q α2 −cos(y1). Ha y1 = 2kπ, akkor ez −α ± √ ( gy , p ), y1 = (2k + 1)π, akkor a sajátértékek −α ± α2 + 1, vagyis az egyensúlyi helyzet bármilyen α mellett instabilis. 5. Bernoulli-féle diﬀerenciálegyenletnek nevezzük az A1(x)y′ + A0(x)y = B(x)yα alakú egyenleteket, ahol α ∈R\{0, 1}, A0, A1 és B adott függvények. Az ilyen egyenleteket u(x) = y(x)1−α helyettesítéssel visszavezethetjük elsőrendű lineáris diﬀerenciálegyenletre. Oldjuk meg az alábbi Bernoulli-féle diﬀerenciálegyenleteket: a) 2y′ + y = (x −1)y3, b) 3(1 + x2)y′ + y = y4. Megoldás. a) α = 3 miatt u(x) = y(x)−2 helyettesítést alkalmazunk. Az új egyenlet u′ = −2 1 3 1 y3y′ = 1 y3 y3 y −(x −1)y3 = 1 y2 −x + 1 = u + 1 −x. = 1 2 A homogén egyenlet megoldása u(x) = Cex, az inhomogén egyenlet megoldása az állandók variálásának módszere szerint u(x) = c(x)ex alakú, ahol c′(x) = e−x(1 −x). Ebből c(x) = xe−x+C, tehát az általános megoldás u(x) = x+Cex. Visszahelyettesítve y(x) = ± 1 √x+Cex. A helyettesítés miatt külön meg kell nézni, hogy mi történik y = 0 esetén. Láthatjuk, hogy ekkor a megoldás azonosan 0, ami az előbbi általános megoldás C →∞limesze. b) Most α = 4, tehát u(x) = y(x)−3 helyettesítés célravezető. Az új egyenlet u′ = −3 1 y4 y −y4 1 y4y′ = 1 y4 y4 y −y4 1 + x2 = 1 y3 y3 1 1 + x2 − 1 1 + x2 = 1 1 + x2u − 1 1 + x2. A homogén egyenlet megoldása u(x) = earctan x, az inhomogéné u(x) = c(x)earctan x, ahol c′(x) = e−arctan x −1 1 + x2, tehát c(x) = e−arctan x + C. Az eredeti egyenlet általános megoldása y(x) = 3q u(x) 3q u 1 3q (e−arctan x + C)earctan x = 1 q (e−arctan x + C)earctan x = 1 3√ 1 + Cearctan x. Most is igaz, hogy y(x) = 0 is megoldás, ami a helyettesítés miatt nem látszik az általános megoldáson.	egyenletrendszert kapjuk, a jobb oldal akkor 0, ha y2 = 0 és − sin(y1) = 0, azaz y1 = kπ valamilyen k ∈ -re. A deriváltmátrix Z � 0 1 _−_ cos(y1) _−2α_ � _,_ _√_ � ennek sajátértékei −α ± _α[2]_ _−_ cos(y1). Ha y1 = 2kπ, akkor ez −α ± _α[2]_ _−_ 1, tehát α > 0 esetén aszimptotikusan stabilis, α = 0 esetén a stabilitás nem dönthető el ezek alapján (egyébként stabilis, de nem aszimptotikusan stabilis), α < 0 esetén instabilis. Ha viszont _√_ _y1 = (2k + 1)π, akkor a sajátértékek −α ±_ _α[2]_ + 1, vagyis az egyensúlyi helyzet bármilyen _α mellett instabilis._ 5. Bernoulli-féle differenciálegyenletnek nevezzük az _A1(x)y[′]_ + A0(x)y = B(x)y[α] alakú egyenleteket, ahol α ∈ R\{0, 1}, A0, A1 és B adott függvények. Az ilyen egyenleteket _u(x) = y(x)[1][−][α]_ helyettesítéssel visszavezethetjük elsőrendű lineáris differenciálegyenletre. Oldjuk meg az alábbi Bernoulli-féle differenciálegyenleteket: a) 2y[′] + y = (x − 1)y[3], b) 3(1 + x[2])y[′] + y = y[4]. _Megoldás._ a) α = 3 miatt u(x) = y(x)[−][2] helyettesítést alkalmazunk. Az új egyenlet _u[′]_ = −2 [1] _y[3]_ _[y][′][ = 1]y[3]_ �y − (x − 1)y[3][�] = [1] _y[2][ −]_ _[x][ + 1 =][ u][ + 1][ −]_ _[x.]_ A homogén egyenlet megoldása u(x) = Ce[x], az inhomogén egyenlet megoldása az állandók variálásának módszere szerint u(x) = c(x)e[x] alakú, ahol c[′](x) = e[−][x](1 − _x)._ Ebből c(x) = xe[−][x]+C, tehát az általános megoldás u(x) = x+Ce[x]. Visszahelyettesítve _y(x) = ±_ _√x+1Cex_ . A helyettesítés miatt külön meg kell nézni, hogy mi történik y = 0 esetén. Láthatjuk, hogy ekkor a megoldás azonosan 0, ami az előbbi általános megoldás _C →∞_ limesze. b) Most α = 4, tehát u(x) = y(x)[−][3] helyettesítés célravezető. Az új egyenlet _y −_ _y[4]_ 1 1 1 1 _u[′]_ = −3 [1] _y[4]_ _[y][′][ = 1]y[4]_ 1 + x[2][ = 1]y[3] 1 + x[2][ −] 1 + x[2][ =] 1 + x[2] _[u][ −]_ 1 + x[2] _[.]_ A homogén egyenlet megoldása u(x) = e[arctan][ x], az inhomogéné u(x) = c(x)e[arctan][ x], ahol _−1_ _c[′](x) = e[−]_ [arctan][ x] 1 + x[2] _[,]_ tehát c(x) = e[−] [arctan][ x] + C. Az eredeti egyenlet általános megoldása 1 1 1 _y(x) =_ = _[√]3_ [�]3 _u(x)_ [�]3 (e[−] [arctan][ x] + C)e[arctan][ x][ =] 1 + Ce[arctan][ x] _[.]_ Most is igaz, hogy y(x) = 0 is megoldás, ami a helyettesítés miatt nem látszik az általános megoldáson. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszert kapjuk, a jobb oldal akkor</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> kπ</span></i></p> <p style="top:73.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">valamilyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> k</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">Z</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re. A deriváltmátrix</span></p> <p style="top:90.8pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:99.4pt;left:132.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:99.4pt;left:177.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:113.9pt;left:112.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:113.9pt;left:168.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i></p> <p style="top:90.8pt;left:191.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:106.7pt;left:199.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:144.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ennek sajátértékei</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:134.0pt;left:202.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:144.4pt;left:212.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">cos(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">kπ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor ez</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:134.7pt;left:429.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:144.4pt;left:439.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span></p> <p style="top:158.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">esetén aszimptotikusan stabilis,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén a stabilitás nem dönthető el ezek alapján</span></p> <p style="top:173.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(egyébként stabilis, de nem aszimptotikusan stabilis),</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α <</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén instabilis. Ha viszont</span></p> <p style="top:187.8pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">k</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor a sajátértékek</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:178.0pt;left:293.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:187.8pt;left:303.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, vagyis az egyensúlyi helyzet bármilyen</span></p> <p style="top:202.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mellett instabilis.</span></p> <p style="top:221.7pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5. Bernoulli-féle diﬀerenciálegyenletnek nevezzük az</span></p> <p style="top:248.1pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></sup></p> <p style="top:274.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú egyenleteket, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\{</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">}</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, A</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> B</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> adott függvények. Az ilyen egyenleteket</span></p> <p style="top:288.9pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">α</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyettesítéssel visszavezethetjük elsőrendű lineáris diﬀerenciálegyenletre.</span></p> <p style="top:303.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Oldjuk meg az alábbi Bernoulli-féle diﬀerenciálegyenleteket:</span></p> <p style="top:317.8pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:334.2pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 3(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:353.7pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:368.1pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> miatt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyettesítést alkalmazunk. Az új egyenlet</span></p> <p style="top:399.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:407.9pt;left:168.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup></p> <p style="top:407.9pt;left:206.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:389.7pt;left:220.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:399.7pt;left:226.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></sup></p> <p style="top:399.7pt;left:303.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:407.9pt;left:317.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1 =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x.</span></i></sup></p> <p style="top:432.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A homogén egyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az inhomogén egyenlet megoldása az</span></p> <p style="top:446.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">állandók variálásának módszere szerint</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">alakú, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:461.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ebből</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ce</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Visszahelyettesítve</span></p> <p style="top:475.4pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></p> <p style="top:473.9pt;left:162.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:477.0pt;left:147.3pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">Ce</span></i><i><span style="font-family:LMMathItalic6,serif;font-size:6.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A helyettesítés miatt külön meg kell nézni, hogy mi történik</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span></p> <p style="top:489.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">esetén. Láthatjuk, hogy ekkor a megoldás azonosan</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami az előbbi általános megoldás</span></p> <p style="top:504.3pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> →∞</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">limesze.</span></p> <p style="top:520.8pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b) Most</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> α</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">helyettesítés célravezető. Az új egyenlet</span></p> <p style="top:554.3pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3 </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:562.5pt;left:168.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup></p> <p style="top:562.5pt;left:206.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:546.2pt;left:219.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:562.5pt;left:219.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 1</span></sup></p> <p style="top:562.5pt;left:269.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:546.2pt;left:295.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:562.5pt;left:282.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:546.2pt;left:344.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:562.5pt;left:331.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:546.2pt;left:394.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:562.5pt;left:381.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup></p> <p style="top:546.2pt;left:449.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:562.5pt;left:436.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:587.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A homogén egyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az inhomogéné</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:601.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ahol</span></p> <p style="top:630.0pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup></p> <p style="top:621.9pt;left:219.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:638.2pt;left:211.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:661.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Az eredeti egyenlet általános megoldása</span></p> <p style="top:692.8pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:684.8pt;left:179.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:704.5pt;left:166.1pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></sup></p> <p style="top:704.1pt;left:176.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:692.8pt;left:203.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:684.8pt;left:274.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:704.5pt;left:216.9pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span><sup><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></sup></p> <p style="top:704.1pt;left:226.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:684.8pt;left:391.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:701.4pt;left:357.4pt;line-height:6.0pt"><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup></p> <p style="top:701.9pt;left:367.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ce</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">arctan</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> x</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:731.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Most is igaz, hogy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> is megoldás, ami a helyettesítés miatt nem látszik az</span></p> <p style="top:745.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">általános megoldáson.</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> </div>	page_323.png	egyenletrendszert kapjuk, a jobb oldal akkor 0, ha 9. valamilyen k € Z-re. A deríváltmátrix sajátértékei —a £ 3? — cos(yi). Ha y, és — sin(y) — 0, azaz yi — kr 2er, akkor ez a £ Va?—1, tehát a — 0 a mellett instabilis. 5. Bernoulli-féle differenciálogyenletnek nevezzük az A(V 4 Aolojy — B" alakú egyenleteket, ahol a € R (0.1), A. A, és B adott függvények. Az ilyen egyenleteket a(z) — v(z)1-e helyettesítéssel visszavezethetjük elsőrendű lincáris dífferenciálegyenletre. Oldjuk meg az alábbi Bernoulli-féle dífferenciálegyenleteket: a) 2/ 4 —(r 1 1449 .Megoldás, a) a — 3 miatt u(z) — y(£)7? helyettes ést alkalmazunk. Az új egyenlet to o] Paloztizutl-s. ot V—A(v (€-1y) F-kádásátái én egyenlet megoldása u(z) — Ce", az inhomogén egyenlet n egoldása az 2(—). Ebből ele) — ze""-C. tehát az általános megoldás u(z) — 23C. Visszahelyettesítve Ah szni, hogy mi történik y — 0 hogy ekkor a megoldás azonosan 0 ami az előbbi általános megoldás állandók variálásának módszere szerint u(z) — elzje? alakú, ahol elr) lyettesítés miatt külön meg kell IVE vr A homogén egyenlet megoldása u(r ahol eee tehát elz) v0- mj AMost is ígaz, hogy y(z) — 0 is megoldás, ami a helyettesítés miatt nem látszik az általános megoldáson.
az általános megoldás u(x) = Ae−2x + Be3x. Innen kétféleképp is folytathatjuk: az eredeti egyenlethez tartozó homogén egyenlet általános megoldását felírva az állandók variálásának módszerével, vagy pedig először az u-ra teljesülő inhomogén egyenletet oldjuk meg, majd ebből helyettesítéssel kapjuk az eredeti egyenlet megoldását. Most az utóbbi módszer gyorsabb: u′′(x) −u′(x) −6u(x) = 5e3x megoldása xe3x + Ae−2x + Be3x, tehát az eredeti egyenlet általános megoldása y(x) = x3 ln x + A 1 x2 + Bx3. 7. Az y′′ = f(y, y′) alakú egyenleteket másodrendű autonóm diﬀerenciálegyenletnek nevezzük. Az ilyen egyenletek megoldásánál érdemes y′(x) = p(y(x)) helyettesítést alkalmazni, ahol tehát y 7→p(y) az új ismeretlen függvény. Ekkor y′′(x) = p′(y(x))y′(x) = p′(y(x))p(y(x)), így behelyettesítés után a pp′ = f(y, p) q elsőrendű diﬀerenciálegyenlethez jutunk. Ennek megoldása után az y′ = p(y) szétválasztható diﬀerenciálegyenletet kell megoldani. Speciális eset: ha f csak az első változótól függ, akkor egy −U primitív függvényt választva (U ′(y) = −f(y, y′)) az első diﬀerenciálegyenlet megoldása p(y) = 2(E −U(y)), ahol az E paraméter neve energia, U(y) pedig a potenciális energia. Oldjuk meg az alábbi másodrendű autonóm diﬀerenciálegyenleteket. a) y′′(1 + y2) = yy′2, b) y′′ = y′2 y y′2 y + y′ √ln y az y(0) = e, y′(0) = 2e kezdeti feltétel mellett, y c) y′′ = 1 y3, d) y′′ = −y. Megoldás. a) y′ = p(y) helyettesítés után az egyenlet pp′(1 + y2) = yp2, ami szétválasztható: Megoldás. p′ p = y 1 + y2, p′ a két oldalt integrálva ln p(y) = ln √1 + y2 + C, tehát p(y) = C1 √1 + y2. A következő lépés y′ = p(y) megoldása. Átrendezve a két oldalt integrálva ln p(y) = ln √1 + y2 + C, tehát p(y) = C1 y′ √1 + y2 = C1, tehát arsinh(y(x)) = C1x + C2, azaz y(x) = sinh(C1x + C2). b) y′ = p(y) helyettesítés után az egyenlet pp′ = p2 y + p √ln y, ami p-vel osztás után lineáris: p′ −1 yp = 1 √ln y.	az általános megoldás u(x) = Ae[−][2][x] + Be[3][x]. Innen kétféleképp is folytathatjuk: az eredeti egyenlethez tartozó homogén egyenlet általános megoldását felírva az állandók variálásának módszerével, vagy pedig először az u-ra teljesülő inhomogén egyenletet oldjuk meg, majd ebből helyettesítéssel kapjuk az eredeti egyenlet megoldását. Most az utóbbi módszer gyorsabb: _u[′′](x) −_ _u[′](x) −_ 6u(x) = 5e[3][x] megoldása xe[3][x] + Ae[−][2][x] + Be[3][x], tehát az eredeti egyenlet általános megoldása _y(x) = x[3]_ ln x + A [1] _x[2][ +][ Bx][3][.]_ 7. Az y[′′] = f (y, y[′]) alakú egyenleteket másodrendű autonóm differenciálegyenletnek nevezzük. Az ilyen egyenletek megoldásánál érdemes y[′](x) = p(y(x)) helyettesítést alkalmazni, ahol tehát y �→ _p(y) az új ismeretlen függvény. Ekkor y[′′](x) = p[′](y(x))y[′](x) = p[′](y(x))p(y(x)),_ így behelyettesítés után a _pp[′]_ = f (y, p) elsőrendű differenciálegyenlethez jutunk. Ennek megoldása után az y[′] = p(y) szétválasztható differenciálegyenletet kell megoldani. Speciális eset: ha f csak az első változótól függ, akkor egy −U primitív függvényt választva (U _[′](y) = −f_ (y, y[′])) az első differenciálegyenlet megoldása p(y) = �2(E − _U_ (y)), ahol az E paraméter neve energia, U (y) pedig a potenciális energia. Oldjuk meg az alábbi másodrendű autonóm differenciálegyenleteket. a) y[′′](1 + y[2]) = yy[′][2], _y[′]_ b) y[′′] = _[y][′][2]_ _√_ _y(0) = e, y[′](0) = 2e kezdeti feltétel mellett,_ _y_ [+] ln y az c) y[′′] = 1 _y[3]_ [,] d) y[′′] = −y. _Megoldás._ a) y[′] = p(y) helyettesítés után az egyenlet pp[′](1 + y[2]) = yp[2], ami szétválasztható: _p[′]_ _y_ _p_ [=] 1 + y[2] _[,]_ _√_ 2 a két oldalt integrálva ln p(y) = ln _[√]1 + y[2]_ + C, tehát p(y) = C1 1 + y . A következő lépés y[′] = p(y) megoldása. Átrendezve _y[′]_ _√_ _C1,_ 2 = 1 + y tehát arsinh(y(x)) = C1x + C2, azaz y(x) = sinh(C1x + C2). b) y[′] = p(y) helyettesítés után az egyenlet _p_ _pp[′]_ = _[p]y[2]_ [+] _√ln y,_ ami p-vel osztás után lineáris: 1 _p[′]_ _−_ [1] _√_ _y_ _[p][ =]_ ln y. -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az általános megoldás</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Innen kétféleképp is folytathatjuk: az</span></p> <p style="top:73.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">eredeti egyenlethez tartozó homogén egyenlet általános megoldását felírva az állandók</span></p> <p style="top:88.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">variálásának módszerével, vagy pedig először az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-ra teljesülő inhomogén egyenletet</span></p> <p style="top:102.4pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">oldjuk meg, majd ebből helyettesítéssel kapjuk az eredeti egyenlet megoldását. Most</span></p> <p style="top:116.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">az utóbbi módszer gyorsabb:</span></p> <p style="top:142.9pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">u</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 5</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup></p> <p style="top:169.0pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> xe</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Ae</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Be</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát az eredeti egyenlet általános megoldása</span></p> <p style="top:199.5pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:207.7pt;left:220.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Bx</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:229.3pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">7. Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakú egyenleteket másodrendű autonóm diﬀerenciálegyenletnek nevezzük.</span></p> <p style="top:243.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Az ilyen egyenletek megoldásánál érdemes</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítést alkalmazni, ahol</span></p> <p style="top:258.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> 7→</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az új ismeretlen függvény. Ekkor</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:272.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">így behelyettesítés után a</span></p> <p style="top:298.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">pp</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></p> <p style="top:324.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">elsőrendű diﬀerenciálegyenlethez jutunk. Ennek megoldása után az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> szétválaszt-</span></p> <p style="top:339.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ható diﬀerenciálegyenletet kell megoldani.</span></p> <p style="top:353.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Speciális eset: ha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> csak az első változótól függ, akkor egy</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> primitív függvényt választva</span></p> <p style="top:369.8pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">f</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y, y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) az első diﬀerenciálegyenlet megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:359.4pt;left:409.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:369.8pt;left:419.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> E</span></i></p> <p style="top:385.3pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">paraméter neve energia,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> pedig a potenciális energia.</span></p> <p style="top:399.7pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Oldjuk meg az alábbi másodrendű autonóm diﬀerenciálegyenleteket.</span></p> <p style="top:414.1pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></p> <p style="top:439.0pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:447.2pt;left:129.6pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:430.9pt;left:165.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:438.3pt;left:156.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel mellett,</span></p> <p style="top:462.6pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:461.1pt;left:128.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:469.9pt;left:125.8pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">3</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:480.3pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:499.5pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:514.0pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítés után az egyenlet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> pp</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> yp</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ami szétválasztható:</span></p> <p style="top:538.3pt;left:128.1pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:554.6pt;left:129.5pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:538.3pt;left:167.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:554.6pt;left:154.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup></p> <p style="top:579.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a két oldalt integrálva</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = ln</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:570.5pt;left:427.0pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:594.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A következő lépés</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> megoldása. Átrendezve</span></p> <p style="top:618.4pt;left:144.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:625.9pt;left:128.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:658.9pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> arsinh(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, azaz</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = sinh(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:675.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> helyettesítés után az egyenlet</span></p> <p style="top:708.5pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">pp</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:716.7pt;left:160.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup></p> <p style="top:700.5pt;left:196.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i></p> <p style="top:707.9pt;left:185.8pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y,</span></i></p> <p style="top:740.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ami</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-vel osztás után lineáris:</span></p> <p style="top:770.8pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup></p> <p style="top:779.0pt;left:151.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup></p> <p style="top:762.7pt;left:192.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:770.1pt;left:181.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y.</span></i></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">4</span></p> </div>	page_325.png	az általános megoldás ulz) — Ae-?" 4. Bet, Innen kétféleképp is folytathatjuk: az oldjuk meg, majd ebből helyettesítéssel kapjuk az eredeti egyenlet megoldását. Most aíg) — aía) — 6ula) — 5" megoldása x 4. Ae""" 4 BEÉT, tehát az eredeti egyenlet általános megoldása. egyenletek megoldásánál érdemes y/(z) — ply(z)) helyett t y.—4 plv) az új ismeretlen függyény. Ekkor y"(z) — plulzjv(z) gy behelyettesítés után a [ Speciális eset: ha / csak az első változótól függ, akkor egy —U primití (U"(g) — —f(.17)) az első differenciálegy egoldása píy) — 2(E — U). ahol az E a) VA FP 2 gyenletnek nevezzük. ítést alkalmazni, ahol ply(e)hplyle). G.p) k megoldása után az y — píy) s alet 9 V/ É é 2 (0) — é, V(0) — 2e kezdeti feltétel melett 94v-$ MY. "Megoldás a) 4 — ply) helyettesítés után az egyenlet pp(14-2) — wp. ami szétválasztható: v 134 a két oldalt integrálya In píy) — In VTF7? 4 C. tehát ply) - C.TF7F. A következő lépés 4/ — píy) megoldása. Átrendezve tehát arsinhíyíz)) — Cr 4. C, azaz y(r) — sinh(Cz 4 C). 1) W — py) helyettesítés után az egyenlet MVI 1. a rE
A homogén egyenlet általános megoldása Cy, az állandók variálásának módszere alapján az inhomogén egyenleté c(y)y, ahol c′(y) = 1 y√ln y, tehát c(y) = 2√ln y + C, p(y) = 2y√ln y + Cy. A kezdeti feltétel alapján 2e = y′(0) = p(y(0)) = p(e), q emiatt C = 0. Végül meg kell oldani az y′(x) = 2y√ln y diﬀerenciálegyenletet y(0) = e kezdeti feltétel mellett. Ez szétválasztható, integrálással kapjuk a megoldást: 2 ln y(x) = 2x, azaz q mellett. Ez szétválasztható, integrálással kapjuk a megoldást: 2 ln y(x) = 2x, azaz y(x) = ex2. c) Ebben az egyenletben nem szerepel y′, U(y) = 1 2y2 jelöléssel y′′ = −U ′(y), amiből y′ = p(y) = q y′ = p(y) = 2(E −U(y)). A szétválasztható egyenletet átrendezzük (y = 0 esetén az egyenlet nem értelmes, tegyük fel, hogy végig y(x) > 0): 1 = y′ q 2(E −U(y)) 1 = y′ q ( y′ E − 1 2y2 = yy′ √2Ey2 −1, mindkét oldalt integrálva x + C = √2Ey2 −1 2E , azaz y(x) = 1 2E + 2E(x + C)2. d) Legyen U(y) = y2 2 , az egyenlet ezzel y′′ = −U ′(y), tehát y′ = √2E −y2. Átrendezzük és mindkét oldalt integráljuk: Legyen U(y) = y2 2 , az egyenlet ezzel y′′ = −U ′(y), tehát y′ = √2E −y2. Átrendezzük és mindkét oldalt integráljuk: y′ √2E −y2 dx = x + C = √ 2E r 1 − √ 2E 2 dy = arcsin y √ 2E . √ Ha rögzítünk egy C értéket, akkor úgy tűnik, mintha x csak a [−π/2 −C, π/2 −C] intervallumban lehetne, tehát a megoldások csak véges sokáig léteznének. Azonban invertálás után az y(x) = 2E sin(x + C) kifejezést kapjuk, ami minden x-re értelmes és meg is oldja a diﬀerenciálegyenletet (ez az általános megoldás). Megjegyzés. Ugyanez történik bármely olyan U(y) és E mellett, amire az {y ∈R\|U(y) ≤ E} halmaz korlátos (és a végpontokon U deriváltja nem 0), ami azzal függ össze, hogy a megoldás ilyenkor periodikus és y′ kétértékű függvénye y-nak. √ 2E sin(x + C) kifejezést kapjuk, ami minden x-re értelmes Ha rögzítünk egy C értéket, akkor úgy tűnik, mintha x csak a [−π/2 −C, π/2 −C] intervallumban lehetne, tehát a megoldások csak véges sokáig léteznének. Azonban invertálás után az y(x) = További gyakorló feladatok 8. Stabilis-e az y′ 1 y′ 1 = −y1 + y3 + y4 y′ 2 = −2y1 + y2 −2 y′ 2 = −2y1 + y2 −2y4 y′ 3 = −y2 + y3 + 2y4 y′ 3 = −y2 + y3 + 2y4 y′ 4 = −2y1 + y2 −y4 y′ 4 = −2y1 + y2 −y4 diﬀerenciálegyenlet-rendszer?	A homogén egyenlet általános megoldása Cy, az állandók variálásának módszere alapján az inhomogén egyenleté c(y)y, ahol c[′](y) = 1 _y[√]ln y_ [, tehát][ c][(][y][) = 2][√][ln][ y][ +][ C][,] _p(y) = 2y[√]ln y + Cy. A kezdeti feltétel alapján_ 2e = y[′](0) = p(y(0)) = p(e), emiatt C = 0. Végül meg kell oldani az y[′](x) = 2y[√]ln y differenciálegyenletet y(0) = e kezdeti feltétel � mellett. Ez szétválasztható, integrálással kapjuk a megoldást: 2 ln y(x) = 2x, azaz _y(x) = e[x][2]._ c) Ebben az egyenletben nem szerepel y[′], U (y) = 1 2y[2][ jelöléssel][ y][′′][ =][ −][U][ ′][(][y][), amiből] � _y[′]_ = p(y) = 2(E − _U_ (y)). A szétválasztható egyenletet átrendezzük (y = 0 esetén az egyenlet nem értelmes, tegyük fel, hogy végig y(x) > 0): _y[′]_ _y[′]_ _yy[′]_ 1 = = _√_ _,_ �2(E − _U_ (y)) �2 �E − 1 � [=] 2Ey2 − 1 2y[2] mindkét oldalt integrálva _√_ 2 2Ey _−_ 1 _x + C =_ _,_ 2E azaz � 1 2E [+ 2][E][(][x][ +][ C][)][2][.] _y(x) =_ d) Legyen U (y) = _[y][2]_ 2 [, az egyenlet ezzel][ y][′′][ =][ −][U][ ′][(][y][), tehát][ y][′][ =][ √][2][E][ −] _[y][2][. Átrendezzük]_ és mindkét oldalt integráljuk: _x + C =_ � _√_ _y[′]_ _x =_ � _√12E_ 2E − _y2 d_ �1 − � _√y_ 2E _y_ _√_ 2 [d][y][ = arcsin] � 2E [.] Ha rögzítünk egy C értéket, akkor úgy tűnik, mintha x csak a [−π/2 − _C, π/2 −_ _C]_ intervallumban lehetne, tehát a megoldások csak véges sokáig léteznének. Azonban _√_ invertálás után az y(x) = 2E sin(x + C) kifejezést kapjuk, ami minden x-re értelmes és meg is oldja a differenciálegyenletet (ez az általános megoldás). _Megjegyzés. Ugyanez történik bármely olyan U_ (y) és E mellett, amire az {y ∈ _\|U_ (y) ≤ R _E} halmaz korlátos (és a végpontokon U deriváltja nem 0), ami azzal függ össze, hogy_ a megoldás ilyenkor periodikus és y[′] kétértékű függvénye y-nak. ## További gyakorló feladatok 8. Stabilis-e az _y1[′]_ [=][ −][y][1] [+][ y][3] [+][ y][4] _y2[′]_ [=][ −][2][y][1] [+][ y][2] _[−]_ [2][y][4] _y3[′]_ [=][ −][y][2] [+][ y][3] [+ 2][y][4] _y4[′]_ [=][ −][2][y][1] [+][ y][2] _[−]_ _[y][4]_ differenciálegyenlet-rendszer? -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A homogén egyenlet általános megoldása</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az állandók variálásának módszere alap-</span></p> <p style="top:73.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ján az inhomogén egyenleté</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ahol</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:72.0pt;left:373.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:81.1pt;left:362.9pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> c</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:90.4pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> Cy</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A kezdeti feltétel alapján</span></p> <p style="top:115.4pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0)) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:140.3pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">emiatt</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:154.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Végül meg kell oldani az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> diﬀerenciálegyenletet</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(0) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kezdeti feltétel</span></p> <p style="top:171.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mellett. Ez szétválasztható, integrálással kapjuk a megoldást:</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span></p> <p style="top:161.1pt;left:433.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:171.5pt;left:443.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">ln</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) = 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, azaz</span></p> <p style="top:189.1pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> e</span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">.</span></p> <p style="top:205.5pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c) Ebben az egyenletben nem szerepel</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:204.0pt;left:360.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:212.8pt;left:356.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> jelöléssel</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span></sup></p> <p style="top:225.0pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> p</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:214.6pt;left:159.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:225.0pt;left:169.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A szétválasztható egyenletet átrendezzük (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> esetén az</span></p> <p style="top:240.5pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenlet nem értelmes, tegyük fel, hogy végig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> ></span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">):</span></p> <p style="top:271.8pt;left:126.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1 =</span></p> <p style="top:263.7pt;left:181.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:272.6pt;left:149.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">q</span></p> <p style="top:283.0pt;left:159.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">))</span></p> <p style="top:271.8pt;left:227.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></p> <p style="top:263.7pt;left:271.2pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:272.6pt;left:241.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">r</span></p> <p style="top:286.0pt;left:251.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:276.0pt;left:259.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:286.0pt;left:265.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:284.5pt;left:294.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:293.3pt;left:290.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:276.0pt;left:304.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup></p> <p style="top:263.7pt;left:349.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">yy</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:271.1pt;left:328.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ey</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:314.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">mindkét oldalt integrálva</span></p> <p style="top:346.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:329.0pt;left:174.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">Ey</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:354.4pt;left:194.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i></p> <p style="top:346.2pt;left:231.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></p> <p style="top:371.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">azaz</span></p> <p style="top:402.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:385.0pt;left:164.6pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">s</span></p> <p style="top:394.6pt;left:180.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:410.9pt;left:175.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:436.2pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d) Legyen</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:443.5pt;left:182.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, az egyenlet ezzel</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000"> ′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> √</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Átrendezzük</span></sup></p> <p style="top:450.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és mindkét oldalt integráljuk:</span></p> <p style="top:484.6pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:472.2pt;left:173.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:476.5pt;left:207.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:484.0pt;left:186.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> d</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:472.2pt;left:268.9pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">Z</span></p> <p style="top:472.8pt;left:313.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:475.8pt;left:306.8pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:482.6pt;left:313.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">E</span></i></p> <p style="top:485.4pt;left:282.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">r</span></p> <p style="top:499.8pt;left:292.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></p> <p style="top:489.8pt;left:312.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:497.9pt;left:326.3pt;line-height:8.0pt"><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></p> <p style="top:501.3pt;left:319.6pt;line-height:8.0pt"><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">√</span></i></p> <p style="top:508.1pt;left:326.7pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">E</span></i></p> <p style="top:489.8pt;left:338.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = arcsin</span></sup></p> <p style="top:476.5pt;left:423.8pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></p> <p style="top:484.0pt;left:414.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:494.1pt;left:424.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E </span></i><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></sup></p> <p style="top:527.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ha rögzítünk egy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> értéket, akkor úgy tűnik, mintha</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> csak a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> [</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C, π/</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">]</span></p> <p style="top:542.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">intervallumban lehetne, tehát a megoldások csak véges sokáig léteznének. Azonban</span></p> <p style="top:556.7pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">invertálás után az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span></p> <p style="top:546.6pt;left:230.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:556.7pt;left:240.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sin(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> C</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> kifejezést kapjuk, ami minden</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> x</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-re értelmes</span></p> <p style="top:571.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">és meg is oldja a diﬀerenciálegyenletet (ez az általános megoldás).</span></p> <p style="top:588.0pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megjegyzés.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Ugyanez történik bármely olyan</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> E</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> mellett, amire az</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> {</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ∈</span></i><span style="font-family:MSBM10,serif;font-size:12.0pt;color:#000000">R</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">\|</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ≤</span></i></p> <p style="top:602.4pt;left:97.7pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">E</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">}</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> halmaz korlátos (és a végpontokon</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> U</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> deriváltja nem</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">), ami azzal függ össze, hogy</span></p> <p style="top:616.8pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a megoldás ilyenkor periodikus és</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kétértékű függvénye</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-nak.</span></p> <p style="top:649.4pt;left:56.7pt;line-height:14.3pt"><b><span style="font-family:LMRoman12,serif;font-size:14.3pt;color:#000000">További gyakorló feladatok</span></b></p> <p style="top:673.5pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">8. Stabilis-e az</span></p> <p style="top:698.4pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:704.5pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:715.8pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:722.0pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:733.3pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:739.4pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:750.7pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:756.8pt;left:112.2pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup></p> <p style="top:775.6pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">diﬀerenciálegyenlet-rendszer?</span></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">5</span></p> </div>	page_326.png	Píu) — 2/ 4 Cy. A kezdeti feltétel alapján 1(0) — plul0)) — ple), emiatt C. Végül me kell oldani az y(2) — 2yy/Iry differenciálegyenletet y(0) — vr e e) Ebben az egyenletben nem szerepel 4 Ufg) — 3lr jelöléssel y" — —U"(y), amiből W — ply) — V2(E — Uty). A szétválasztható egyenletet átrendezzük (y — 0 eset egyenlet nem értelmes, tegyük fel, hogy vésig vlz) — 0). 1 UA VAE-UGy mindkét oldalt integrálva V33 2ő4 C. Y — VZEZ. Átrendezzük a) Legyen Uly) — 2£. az egyenlet eszel 4" és mindkét oldalt integráljuk: 24C- E / 1- V( Ha rögzítünk egy C értéket, akkor úgy tűnik, mintha 2 csak a [-r/2 — C.2/2- CI AMegjegyzés. Ugyanez történik bármely olyan U(y) és £ mellett, amire az (y € RIU(y) £ ÁZ dy — arcsin "További gyakorló feladatok 8. Stabilis-e az DERVETET] W 2n 492 — 294 1ú v 4 99 4291 W 2m t9 differenciálegyer dszer?
Megoldás. Az egyenletrendszer lineáris, y′ = Ay alakú, ahol y = (y1, y2, y3, y4) és      −1 0 1 1 −2 1 0 −2 0 −1 1 2 −2 1 0 −1 2 . A = A karakterisztikus egyenlet 0 = det(A −λI) = λ4 + 2λ2 + 1 = (1 + λ2)2, tehát λ = ±i a két gyök, mindkettő kétszeres. Az i-hez tartozó sajátvektorokat a      −1 −i 0 1 1 −2 1 −i 0 −2 0 −1 1 −i 2 −2 1 0 −1 −i ·  x x x     x1 x2 x3 x4 = 0 egyenletrendszer megoldásai adják, ezek (0, −1 −i, 1, −1) többszörösei. Mivel ezek nem kétdimenziós alteret alkotnak, az i sajátértékhez egy darab 2 × 2 Jordan-blokk tartozik, tehát az egyenletrendszer instabilis. 9. Határozzuk meg az alábbi diﬀerenciálegyenlet-rendszerek stacionárius pontjait. Milyen típusúak stabilitás tekintetében? a) y′ 2 1 = −2y1 + y3 y′ 2 = −2y1 + y2 b) y′ 1 y′ 1 1 = y2 y′ 1 = y2 1 + y2 −y1y2 −2 y′ 1 −2y2 + y1y2 + 2 = −y2 y′ 1 2 = −y2 y2 1 −2y2 + y1y2 + 4 c) y′ 1 y′ 1 = sin y1 + sin y2 y′ 2 = −sin y1 + sin y′ 2 = −sin y1 + sin y2 d) y′ 1 y′ 1 1 = (1 −y2 y2 2 1 −y2 y′ 1 −y2 1 = (1 −y2 2)y1 y′ 2)y2 1 −y2 2 = (1 −y2 y′ 1 2 = (1 −y2 y2 2 1 −y2 y2 2)y2 e) y′ 1 y′ 2 1 = (y1 −1)2 + y2 y′ 1 = (y1 −1)2 + y2 2 −2 y′ 2 −2 2 = (y1 + 1)2 + y2 y1 (y ) y2 y′ 2 2 = (y1 + 1)2 + y2 y2 2 −2 Megoldás. a) A stacionárius pontok a 0 = −2y1 + y3 2 A stacionárius pontok a 0 = −2y1 + y3 2 = −2y1 + y2 egyenletrendszer megoldásai. A két egyenlet különbsége 0 = y3 2 −y2 = y2(y2 2 −1) = y2(y2 −1)(y2 + 1), amiből y2 = 0 y3 2 −y2 = y2(y2 2 l ből d két egyenlet különbsége 0 = y3 2 −y2 = y2(y2 2 −1) = y2(y2 −1)(y2 + 1), amiből y2 = 0 vagy y2 = ±1, a második egyenletből pedig y1 = y2 2 . A jobb oldal deriváltmátrixa y2 2 . A jobb oldal deriváltmátrixa D(y1,y2) = " # −2 3y2 . 2 −2 1 Ennek karakterisztikus polinomja det(D(y1,y2) −λI) = λ2 + λ + 6y2 2 Ennek karakterisztikus polinomja det(D(y1,y2) −λI) = λ2 + λ + 6y2 √ 2 −2. Ha y2 = 0, akkor a gyökök −2 és 1, ha pedig y2 = ±1, akkor 1 2(−1 ± i 15), tehát (0, 0) instabilis, 1 1 √ akkor a gyökök −2 és 1, ha pedig y2 = ±1, akkor 1 2(−1 ± i 15), tehát (0, 0) instabilis, (1, 1 2) és (−1, −1 2) aszimptotikusan stabilis egyensúlyi pontok. 1 2(−1 ± i úl 1 2) és (−1, −1 2 1 2) aszimptotikusan stabilis egyensúlyi pontok.	_Megoldás. Az egyenletrendszer lineáris, y[′]_ = Ay alakú, ahol y = (y1, y2, y3, y4) és _A =_ −1 0 1 1  _−2_ 1 0 _−2_ _._ 0 _−1_ 1 2   _−2_ 1 0 _−1_ A karakterisztikus egyenlet 0 = det(A − _λI) = λ[4]_ + 2λ[2] + 1 = (1 + λ[2])[2], tehát λ = ±i a két gyök, mindkettő kétszeres. Az i-hez tartozó sajátvektorokat a −1 − _i_ 0 1 1  x1 _−2_ 1 − _i_ 0 _−2_ _·_ _x2_ = 0 0 _−1_ 1 − _i_ 2 _x3_     _−2_ 1 0 _−1 −_ _i_ _x4_ egyenletrendszer megoldásai adják, ezek (0, −1 − _i, 1, −1) többszörösei. Mivel ezek nem_ kétdimenziós alteret alkotnak, az i sajátértékhez egy darab 2 × 2 Jordan-blokk tartozik, tehát az egyenletrendszer instabilis. 9. Határozzuk meg az alábbi differenciálegyenlet-rendszerek stacionárius pontjait. Milyen típusúak stabilitás tekintetében? a) _y1[′]_ [=][ −][2][y][1] [+][ y]2[3] _y2[′]_ [=][ −][2][y][1] [+][ y][2] b) _y1[′]_ [=][ y]1[2] [+][ y][2] _[−]_ _[y][1][y][2]_ _[−]_ [2] _y2[′]_ [=][ −][y]1[2] _[−]_ [2][y][2] [+][ y][1][y][2] [+ 4] c) _y1[′]_ [= sin][ y][1] [+ sin][ y][2] _y2[′]_ [=][ −] [sin][ y][1] [+ sin][ y][2] d) _y1[′]_ [= (1][ −] _[y]1[2]_ _[−]_ _[y]2[2][)][y][1]_ _y2[′]_ [= (1][ −] _[y]1[2]_ _[−]_ _[y]2[2][)][y][2]_ e) _y1[′]_ [= (][y][1] _[−]_ [1)][2][ +][ y]2[2] _[−]_ [2] _y2[′]_ [= (][y][1] [+ 1)][2][ +][ y]2[2] _[−]_ [2] _Megoldás._ a) A stacionárius pontok a 0 = −2y1 + y2[3] [=][ −][2][y][1] [+][ y][2] [egyenletrendszer megoldásai. A] két egyenlet különbsége 0 = y2[3] _[−]_ _[y][2]_ [=][ y][2][(][y]2[2] _[−]_ [1) =][ y][2][(][y][2] _[−]_ [1)(][y][2] [+ 1), amiből][ y][2] [= 0] vagy y2 = ±1, a második egyenletből pedig y1 = _[y]2[2]_ [. A jobb oldal deriváltmátrixa] � _._ _D(y1,y2) =_ �−2 3y2[2] _−2_ 1 Ennek karakterisztikus polinomja det(D(y1,y2) − _λI) = λ[2]_ _√+ λ + 6y2[2]_ _[−]_ [2. Ha][ y][2] [= 0,] akkor a gyökök −2 és 1, ha pedig y2 = ±1, akkor 2[1] [(][−][1][ ±][ i] 15), tehát (0, 0) instabilis, (1, [1] 2 [) és (][−][1][,][ −] [1]2 [) aszimptotikusan stabilis egyensúlyi pontok.] -----	<div id="page0" style="width:595.3pt;height:841.9pt"> <p style="top:59.1pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Az egyenletrendszer lineáris,</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> A</span></i><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> alakú, ahol</span><b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> y</span></b><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> = (</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">, y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></p> <p style="top:103.2pt;left:106.4pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:75.3pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:92.9pt;left:131.0pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:81.5pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:81.5pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:81.5pt;left:187.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:81.5pt;left:208.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:95.9pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:95.9pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:95.9pt;left:187.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:95.9pt;left:203.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:110.4pt;left:142.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:110.4pt;left:162.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:110.4pt;left:187.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:110.4pt;left:208.3pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:124.8pt;left:137.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:124.8pt;left:167.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:124.8pt;left:187.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:124.8pt;left:203.7pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:75.3pt;left:218.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:92.9pt;left:218.8pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:147.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">A karakterisztikus egyenlet</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 = det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">A</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1 = (1 +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> a</span></p> <p style="top:162.4pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">két gyök, mindkettő kétszeres. Az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">-hez tartozó sajátvektorokat a</span></p> <p style="top:178.0pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:195.5pt;left:106.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:184.1pt;left:113.1pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i></p> <p style="top:184.1pt;left:166.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:184.1pt;left:200.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:184.1pt;left:239.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:198.5pt;left:122.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:198.5pt;left:156.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i></p> <p style="top:198.5pt;left:200.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:198.5pt;left:234.9pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:213.0pt;left:127.0pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:213.0pt;left:161.5pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:213.0pt;left:191.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i></p> <p style="top:213.0pt;left:239.6pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:227.4pt;left:122.4pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:227.4pt;left:166.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:227.4pt;left:200.5pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">0</span></p> <p style="top:227.4pt;left:225.6pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i</span></i></p> <p style="top:178.0pt;left:259.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:195.5pt;left:259.4pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">·</span></i></p> <p style="top:178.0pt;left:274.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:195.5pt;left:274.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:184.1pt;left:281.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></p> <p style="top:198.5pt;left:281.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:213.0pt;left:281.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></p> <p style="top:227.4pt;left:281.3pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">x</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">4</span></p> <p style="top:178.0pt;left:292.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span></p> <p style="top:195.5pt;left:292.7pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000"></span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></p> <p style="top:250.0pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszer megoldásai adják, ezek</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">i,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> többszörösei. Mivel ezek nem</span></p> <p style="top:264.5pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">kétdimenziós alteret alkotnak, az</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> sajátértékhez egy darab</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ×</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> Jordan-blokk tartozik,</span></p> <p style="top:278.9pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">tehát az egyenletrendszer instabilis.</span></p> <p style="top:296.8pt;left:62.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">9. Határozzuk meg az alábbi diﬀerenciálegyenlet-rendszerek stacionárius pontjait.</span></p> <p style="top:296.8pt;left:503.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Milyen</span></p> <p style="top:311.2pt;left:77.2pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">típusúak stabilitás tekintetében?</span></p> <p style="top:325.7pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a)</span></p> <p style="top:348.4pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:354.5pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:354.5pt;left:198.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:365.8pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:372.0pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:388.6pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">b)</span></p> <p style="top:411.3pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:417.5pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:417.5pt;left:158.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:428.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:434.9pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:434.9pt;left:168.1pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 4</span></sup></p> <p style="top:451.5pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">c)</span></p> <p style="top:474.2pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:480.4pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:491.7pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:497.8pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ sin</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:514.4pt;left:80.8pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">d)</span></p> <p style="top:537.1pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:543.3pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:543.3pt;left:183.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:543.3pt;left:209.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:554.6pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:560.7pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:560.7pt;left:183.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:560.7pt;left:209.3pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:577.3pt;left:82.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">e)</span></p> <p style="top:600.1pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:606.2pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:606.2pt;left:218.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:617.5pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><i><span style="font-family:LMMathSymbols8,serif;font-size:8.0pt;color:#000000">′</span></i></sup></p> <p style="top:623.6pt;left:132.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= (</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1)</span></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:623.6pt;left:217.8pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup></p> <p style="top:640.2pt;left:77.2pt;line-height:12.0pt"><i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Megoldás.</span></i></p> <p style="top:654.7pt;left:81.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">a) A stacionárius pontok a</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> +</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:660.8pt;left:296.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">egyenletrendszer megoldásai. A</span></sup></p> <p style="top:669.1pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">két egyenlet különbsége</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0 =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">3</span></sup></p> <p style="top:675.3pt;left:251.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">=</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:675.3pt;left:319.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1) =</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1)(</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+ 1)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, amiből</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></sup></p> <p style="top:683.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">vagy</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, a második egyenletből pedig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i></sup><sup><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span></sup></p> <p style="top:690.9pt;left:351.9pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. A jobb oldal deriváltmátrixa</span></sup></p> <p style="top:714.9pt;left:126.9pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span></p> <p style="top:699.0pt;left:178.5pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">"</span></p> <p style="top:707.6pt;left:184.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:707.6pt;left:209.4pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">3</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:713.7pt;left:221.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></p> <p style="top:722.0pt;left:184.3pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></p> <p style="top:722.0pt;left:214.9pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></p> <p style="top:699.0pt;left:226.2pt;line-height:10.0pt"><span style="font-family:LMMathExtension10,serif;font-size:10.0pt;color:#000000">#</span></p> <p style="top:714.9pt;left:234.0pt;line-height:12.0pt"><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">.</span></i></p> <p style="top:745.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">Ennek karakterisztikus polinomja</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> det(</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">D</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">(</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">1</span><i><span style="font-family:LMMathItalic8,serif;font-size:8.0pt;color:#000000">,y</span></i><span style="font-family:LMRoman6,serif;font-size:6.0pt;color:#000000">2</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">)</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">λI</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">) =</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">+</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> λ</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> + 6</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">y</span></i><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup></p> <p style="top:751.8pt;left:444.4pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> </span></i><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">. Ha</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span></sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">= 0</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">,</span></sup></p> <p style="top:761.2pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">akkor a gyökök</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, ha pedig</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> y</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> =</span><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, akkor</span><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:768.5pt;left:353.0pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> ±</span></i></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000"> i</span></i></sup></p> <p style="top:751.4pt;left:396.2pt;line-height:12.0pt"><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">√</span></i></p> <p style="top:761.2pt;left:406.1pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">15)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">, tehát</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (0</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> 0)</span><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> instabilis,</span></p> <p style="top:775.6pt;left:97.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">(1</span><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000"> </span><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:782.9pt;left:114.5pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> és</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> (</span></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000">−</span></i></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">1</span></sup><sup><i><span style="font-family:LMMathItalic12,serif;font-size:12.0pt;color:#000000">,</span></i></sup><sup><i><span style="font-family:LMMathSymbols10,serif;font-size:12.0pt;color:#000000"> −</span></i></sup><sup><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">1</span></sup></p> <p style="top:782.9pt;left:177.6pt;line-height:8.0pt"><span style="font-family:LMRoman8,serif;font-size:8.0pt;color:#000000">2</span><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">)</span></sup><sup><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000"> aszimptotikusan stabilis egyensúlyi pontok.</span></sup></p> <p style="top:805.5pt;left:294.7pt;line-height:12.0pt"><span style="font-family:LMRoman12,serif;font-size:12.0pt;color:#000000">6</span></p> </div>	page_327.png	.Megoldás. Az egyen Ay alakú, ahol y — (y1.92.95-94) és A karakterisztikus egyenlet 0 — det(A — A) két gyök, mindket 944222 41 (14- 2J, tehát ) — ti a ő kétszeres. Az i-hez tartozó sajátvektorokat a 1-i 0 1 1 ] [rr 2 1-i 0 22 0 - 1-i 2 [ -2 1 0 -1 ll egyenletrendszer megoldásai adják, ezek (0,—1 — 1. 1,—1) többszörösei. . Mivel ezek kétdimenziós alteret alkotnak, az : sajátértékhez egy darab 2 x 2 Jordan-blokk tartozik. tel ít az egyenletrendszer instabilis. Határozzuk meg az alábbi differenciálogyenlet-rendszerek stacionárius pontjait. . Milyen típusúak stabilítás tekintetében? a úo 2m t9 " Wt 2-n 2nn t1 9 sinyi n 4 ú-0-ú-Ön 0 e 9 méé -2 ( 2 "Megoldás a) A stacionárius pontok a 0 — —2y, 4 ] — —2y1 4 9. egyenletrendszer megoldásai. A két egyenlet különbsége 0. vagy 9 2 342 Ennek karakterisztikus polinomja det(Díy, yg — AD) — 32 4X 4 6y$ — 2. Ha ya — 0. akkor a győkök —2 és 1, ha pedig a — 4:1, akkor 2(—1 -: 44/15), tehát (0.0) instabilis (1.1) és (-1,—1) aszímptotikusan stabilis egyensúlyi pontok. 1— 2 — 2l8 — 1) — yalya — I1(ye 4 1), amiből ya — 6
A műnek erre a változatára a Nevezd meg! – Így add tovább! 3.0 licenc feltételei1 érvényesek. A következőket teheted a művel: szabadon másolhatod, terjesztheted, bemutathatod és előadhatod a művet származékos műveket (feldolgozásokat) hozhatsz létre kereskedelmi célra is felhasználhatod a művet Az alábbi feltételekkel: Nevezd meg! – A szerző vagy a jogosult által meghatározott módon fel kell tüntet- ned a műhöz kapcsolódó információkat (pl. a szerző nevét vagy álnevét, a Mű címét). Így add tovább! – Ha megváltoztatod, átalakítod, feldolgozod ezt a művet, az így létrejött alkotást csak a jelenlegivel megegyező licenc alatt terjesztheted. A szerző a lehetőségei szerinti legnagyobb gondossággal járt el a könyv írása közben. De ettől még számítani kell szerkesztői pontatlanságra, sőt nem zárható ki a tárgyi tévedés sem. Mindezzel együtt a könyv alkalmas az alapismeretek megszerzésére. A kiadvány létrejöttét az FSF.hu Alapítvány2 támogatta. Szakmai lektor: Palócz István3 Nyelvi lektor: Kauka Béla és Nagyné Kauka Adrienn A könyv elektronikus változata elérhető a http://nagygusztav.hu/ oldalról. 1 http://creativecommons.org/licenses/by-sa/3.0/deed.hu 2 http://fsf.hu/ 3 http://palocz.hu/	A műnek erre a változatára a Nevezd meg! – Így add tovább! 3.0 licenc feltételei[1] érvényesek. #### A következőket teheted a művel:  szabadon másolhatod, terjesztheted, bemutathatod és előadhatod a művet  származékos műveket (feldolgozásokat) hozhatsz létre  kereskedelmi célra is felhasználhatod a művet #### Az alábbi feltételekkel:  Nevezd meg! – A szerző vagy a jogosult által meghatározott módon fel kell tüntet ned a műhöz kapcsolódó információkat (pl. a szerző nevét vagy álnevét, a Mű címét).  Így add tovább! – Ha megváltoztatod, átalakítod, feldolgozod ezt a művet, az így létrejött alkotást csak a jelenlegivel megegyező licenc alatt terjesztheted. A szerző a lehetőségei szerinti legnagyobb gondossággal járt el a könyv írása közben. De ettől még számítani kell szerkesztői pontatlanságra, sőt nem zárható ki a tárgyi tévedés sem. Mindezzel együtt a könyv alkalmas az alapismeretek megszerzésére. -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:136.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A műnek erre a változatára a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Nevezd meg! – Így add tovább! 3.0</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> licenc feltételei</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">1</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> érvénye-</span></p> <p style="top:148.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">sek.</span></p> <p style="top:170.7pt;left:85.2pt;line-height:13.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:13.0pt;color:#000000">A következőket teheted a művel:</span></p> <p style="top:196.9pt;left:99.4pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szabadon másolhatod, terjesztheted, bemutathatod és előadhatod a művet</span></p> <p style="top:217.2pt;left:99.4pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">származékos műveket (feldolgozásokat) hozhatsz létre</span></p> <p style="top:237.4pt;left:99.4pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kereskedelmi célra is felhasználhatod a művet </span></p> <p style="top:259.2pt;left:85.2pt;line-height:13.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:13.0pt;color:#000000">Az alábbi feltételekkel:</span></p> <p style="top:285.4pt;left:99.4pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Nevezd meg! – A szerző vagy a jogosult által meghatározott módon fel kell tüntet-</span></p> <p style="top:298.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ned a műhöz kapcsolódó információkat (pl. a szerző nevét vagy álnevét, a Mű cí-</span></p> <p style="top:310.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">mét).</span></p> <p style="top:330.8pt;left:99.4pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Így add tovább! – Ha megváltoztatod, átalakítod, feldolgozod ezt a művet, az így </span></p> <p style="top:343.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">létrejött alkotást csak a jelenlegivel megegyező licenc alatt terjesztheted. </span></p> <p style="top:380.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A szerző a lehetőségei szerinti legnagyobb gondossággal járt el a könyv írása közben. De </span></p> <p style="top:393.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ettől még számítani kell szerkesztői pontatlanságra, sőt nem zárható ki a tárgyi tévedés </span></p> <p style="top:405.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">sem. Mindezzel együtt a könyv alkalmas az alapismeretek megszerzésére.</span></p> <p style="top:443.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A kiadvány létrejöttét az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">FSF.hu Alapítvány</span></i><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">2</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> támogatta.</span></p> <img style="position:absolute;transform:matrix(.58177778,0,-0,.5960318,50.733324,580.8667)" src="data:image/png;base64, iVBORw0KGgoAAAANSUhEUgAAASwAAACoCAIAAADVSURYAAAACXBIWXMAAA7EAAAO xAGVKw4bAAAdbklEQVR4nO2dC1iUxf7HAzXN7KIplopWYumpTMssPf86mR0tj55M 89jpfjPTp7Inu5hZVsdOac+pFFjeRe53BQQFEfAKeANEERW5CCJ3QQQVkPv+v/uO +/qyNxdcdtj19z5ffZbdeeedmXc+8/vNvPPO3HSTziHIpCSRSNctiSZBlzct9qQT 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style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">http://nagygusztav.hu/</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> oldalról.</span></p> <p style="top:698.6pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">1</span></p> <p style="top:699.4pt;left:99.4pt;line-height:8.0pt"><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c">http://creativecommons.org/licenses/by-sa/3.0/deed.hu</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:708.8pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">2</span></p> <p style="top:709.6pt;left:99.4pt;line-height:8.0pt"><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c">http://fsf.hu/</span><b><span style="font-family:MagyarLinLibertineB,serif;font-size:9.0pt;color:#000000"> </span></b></p> <p style="top:719.1pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">3</span></p> <p style="top:719.9pt;left:99.4pt;line-height:8.0pt"><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c">http://palocz.hu/</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> </div>	page_330.png	a Nevezd meg! - Így add továbbt 3.0 icenc feltételet érvénye. A következőket teheted a művel: 1 szabadon másolhatod, terjesztheted, bemutathatod és előadhatod a művet a . származékos műveket (feldolgozásokat) hozhatsz létre. M kereskedelmi célra is felhasználhatod a művet Az alábbi feltételekkel: M Nevezd meg! - A szerző vagy a jogosult által meghatározott módon fel kell tüntet. med a műhöz kapcsolódó információkat (pl. a szerző nevét vagy álnevét, a Mű ci. mén,. m Így add tovább! - Ha megváltoztatod, átalakítod, feldolgozod ezt a művet, a; tétrejött alkotást csak a jelenlegível megegyező licenc alatt terjesztheted. A szerző a lehetőségei szerinti legnagyobb gondossággal járt el a könyv írása közben. De ettől még számítani kell szerkesztői pontatlanságra, sőt nem zárható ki a tárgyi tévedés sem. Mindezzel együtt a könyv alkalmas az alapismeretek megszerzésére. A kiadvány létrejöttét az FSF.u Alapítványi támogatta. SSzakmai lektor: Palócz István? Nyelvi lektor: Kauka Béla és Nagyné Kauka Adrienn A könyv elektronikus változata elérhető a htto//nagygusztahu/ oldalról. — ] 2. szpífinal 3 kepíjsssztel
36. oldal 2. A Drupal felhasználói szemmel 2.1. Mi a Drupal? Néhány alapfogalmat érdemes tisztázni a Drupallal kapcsolatban is. 2.1.1. A Drupal felépítése Drupal oldalunk építésekor a CMS motor központi mag része (core) és a kiegészítők (contributions) között különbséget kell tennünk. Drupal Motor A Drupal alapfunkcionalitásait megvalósító alkalmazás. Már önmagában is rendkívül sok szolgáltatással bír, mégis alapvetően az a feladata, hogy a különböző funkciókat hatékonyan fogja össze. Garantált, hogy az itt található kódok alaposan teszteltek, az esetek döntő többségében korrektek és használhatóak, valamint a Drupal alapkoncepciójához illeszkednek. A felfedett hibákra igen gyorsan újabb kiadással reagálnak. Kiegészítők A Drupal közösség által beküldött kiegészítő funkcionalitások (modulok), kinézetek (sminkek), felületfordítások és dokumentációk tartoznak ide. Jellegénél fogva nincs olyan erős irányítás alatt, mint a motor, ezért nem csak tökéletesen működő komponenseket találhatunk itt. Sajnos előfordul, hogy a kiegészítők fejlesztője egy idő után már nem tartja karban a projektjét. Körültekintéssel kell tehát a kiegészítőket használnunk. Másrészt tudnunk kell, hogy a fejlesztők (mind a mag, mind a kiegészítők esetén) megkülönböztetnek stabil és fejlesztői (dev jelöléssel ellátott) változatot. Az utóbbiakat csak óvatosan, nagy körültekintéssel szabad használni. (Tanuláshoz esetleg alkalmazhatók, de éles környezetben inkább korábbi, stabil változatot használjunk.) 2.1.2. Ingyenes a Drupal? A Drupal azon kívül, hogy sok ember szabadidejébe került, nagyon sok cég fizetett munkásainak fizetett munkáját tartalmazza. Pont ez a lényeg, hogy az OpenSource nem egy gittegyelet, amit próbálnak sulykolni egyes ellenérdekelt cégek, hanem egy üzleti modell (non és for profit) aminek keretében a program forráskódja nem zárt, hanem mindenki számára elérhető. Az Examiner, Acqua, Lullabot, de akár a magyar cégek, mint pl. az IntegralVision40 is jelentős munkaidőd áldoz a Drupal fejlesztésére a kódok elérhetővé tételére. 40 http://integralvision.hu/	##### 36. oldal 2. A Drupal felhasználói szemmel ## 2.1. Mi a Drupal? Néhány alapfogalmat érdemes tisztázni a Drupallal kapcsolatban is. ### 2.1.1. A Drupal felépítése Drupal oldalunk építésekor a CMS motor központi mag része (core) és a kiegészítők (cont_ributions) között különbséget kell tennünk._ #### Drupal Motor A Drupal alapfunkcionalitásait megvalósító alkalmazás. Már önmagában is rendkívül sok szolgáltatással bír, mégis alapvetően az a feladata, hogy a különböző funkciókat hatékonyan fogja össze. Garantált, hogy az itt található kódok alaposan teszteltek, az esetek döntő többségében korrektek és használhatóak, valamint a Drupal alapkoncepciójához illeszkednek. A felfedett hibákra igen gyorsan újabb kiadással reagálnak. #### Kiegészítők A Drupal közösség által beküldött kiegészítő funkcionalitások (modulok), kinézetek (sminkek), felületfordítások és dokumentációk tartoznak ide. Jellegénél fogva nincs olyan erős irányítás alatt, mint a motor, ezért nem csak tökéletesen működő komponenseket találhatunk itt. Sajnos előfordul, hogy a kiegészítők fejlesztője egy idő után már nem tartja karban a projektjét. Körültekintéssel kell tehát a kiegészítőket használnunk. Másrészt tudnunk kell, hogy a fejlesztők (mind a mag, mind a kiegészítők esetén) megkülönböztetnek stabil és fejlesztői (dev jelöléssel ellátott) változatot. Az utóbbiakat csak óvatosan, nagy körültekintéssel szabad használni. (Tanuláshoz esetleg alkalmazhatók, de éles környezetben inkább korábbi, stabil változatot használjunk.) ### 2.1.2. Ingyenes a Drupal? A Drupal azon kívül, hogy sok ember szabadidejébe került, nagyon sok cég fizetett munkásainak fizetett munkáját tartalmazza. Pont ez a lényeg, hogy az OpenSource nem egy gittegyelet, amit próbálnak sulykolni egyes ellenérdekelt cégek, hanem egy üzleti modell (non és for profit) aminek keretében a program forráskódja nem zárt, hanem mindenki számára elérhető. Az Examiner, Acqua, Lullabot, de akár a magyar cégek, mint pl. az IntegralVision[40] is jelentős munkaidőd áldoz a Drupal fejlesztésére a kódok elérhetővé tételére. 40 http://integralvision.hu/ -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">36. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:166.0pt;left:85.2pt;line-height:20.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:20.0pt;color:#000000">2.1. Mi a Drupal?</span></b></p> <p style="top:207.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Néhány alapfogalmat érdemes tisztázni a Drupallal kapcsolatban is.</span></p> <p style="top:235.0pt;left:85.2pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.1.1. A Drupal felépítése</span></b></p> <p style="top:267.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Drupal oldalunk építésekor a CMS motor központi mag része (</span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">core</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">) és a kiegészítők (</span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">cont-</span></i></p> <p style="top:280.0pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">ributions</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">) között különbséget kell tennünk.</span></p> <p style="top:301.8pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Drupal Motor</span></b></p> <p style="top:326.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A Drupal alapfunkcionalitásait megvalósító alkalmazás. Már önmagában is rendkívül sok </span></p> <p style="top:339.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szolgáltatással bír, mégis alapvetően az a feladata, hogy a különböző funkciókat hatéko-</span></p> <p style="top:351.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nyan fogja össze. Garantált, hogy az itt található kódok alaposan teszteltek, az esetek dön-</span></p> <p style="top:364.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tő többségében korrektek és használhatóak, valamint a Drupal alapkoncepciójához illesz-</span></p> <p style="top:376.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kednek. A felfedett hibákra igen gyorsan újabb kiadással reagálnak.</span></p> <p style="top:398.5pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Kiegészítők</span></b></p> <p style="top:423.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A Drupal közösség által beküldött kiegészítő funkcionalitások (modulok), kinézetek (smin-</span></p> <p style="top:435.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kek), felületfordítások és dokumentációk tartoznak ide. Jellegénél fogva nincs olyan erős </span></p> <p style="top:448.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">irányítás alatt, mint a motor, ezért nem csak tökéletesen működő komponenseket találha-</span></p> <p style="top:460.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tunk itt. Sajnos előfordul, hogy a kiegészítők fejlesztője egy idő után már nem tartja kar-</span></p> <p style="top:473.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ban a projektjét.</span></p> <p style="top:492.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Körültekintéssel kell tehát a kiegészítőket használnunk.</span></p> <p style="top:510.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Másrészt tudnunk kell, hogy a fejlesztők (mind a mag, mind a kiegészítők esetén) megkü-</span></p> <p style="top:523.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lönböztetnek stabil és fejlesztői (</span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">dev</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> jelöléssel ellátott) változatot. Az utóbbiakat csak óva-</span></p> <p style="top:536.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tosan, nagy körültekintéssel szabad használni. (Tanuláshoz esetleg alkalmazhatók, de éles </span></p> <p style="top:548.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">környezetben inkább korábbi, stabil változatot használjunk.)</span></p> <p style="top:576.5pt;left:85.2pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.1.2. Ingyenes a Drupal?</span></b></p> <p style="top:609.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A Drupal azon kívül, hogy sok ember szabadidejébe került, nagyon sok cég fizetett mun-</span></p> <p style="top:621.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kásainak fizetett munkáját tartalmazza. Pont ez a lényeg, hogy az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">OpenSource</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> nem egy </span></p> <p style="top:634.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">gittegyelet, amit próbálnak sulykolni egyes ellenérdekelt cégek, hanem egy üzleti modell </span></p> <p style="top:646.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(non és for profit) aminek keretében a program forráskódja nem zárt, hanem mindenki </span></p> <p style="top:659.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">számára elérhető.</span></p> <p style="top:677.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Examiner</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">, </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Acqua</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">, </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Lullabot</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">, de akár a magyar cégek, mint pl. az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">IntegralVision</span></i><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">40</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> is je-</span></p> <p style="top:690.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lentős munkaidőd áldoz a Drupal fejlesztésére a kódok elérhetővé tételére.</span></p> <p style="top:719.1pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">40</span><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c"> http://integralvision.hu/</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> </div>	page_331.png	36. oldal 2. A Drupal felhasználói szemmel 1.Mia Drueal? Néhány alapfogalmat érdemes tisztázni a Drupallal kapcsolatban ís. 2.1.1. A Drupal felépítése Drupal oldalunk építésekor a CMS motor központi mag része (core) és a kiegészítők (cont- vibutions) között különbséget kell tennünk. Drupal Motor A Drupal alapfunkcionalitásait megvalósító alkalmazás. Már önmagában is rendkávül sok. szolgáltatással bír, mégis alapvetően az a feladata, hogy a különböző funkciókat hatéko. ító kódok alaposan teszteltek, az esetek dön. éktek és használhatóak, valamint a Drupal alapkoncepciójához illesz. kednek. A felfedett hibákra igen gyorsan újabb kiadással reagálnak. Kiegészítők A Drupal közösség által beküldött kiegészítő funkcionalítások (modulok), kinézetek (smin. kek), felületfordítások és dokumentációk tartoznak ide. Jellegénél fogva nincs olyan erős irányítás alatt, mint a motor, ezért nem csak tökéletesen működő komponenseket találha: tunk átt. Sajnos előfordul, hogy a kiegészítők fejlesztője egy ídő után már nem tartja kar. ban a projektjét. rültekintéssel kell tehát a kiegészítőket használnunk. zészt tudnunk kell, hogy a fejlesztők (mind a mag, mind a kiegészítők esetén) megkü. tönböztetnek stabil és fejlesztői (dev jelöléssel ellátott) változatot. Az utóbbiakat csak óva- tosan, nagy körültekintéssel szabad használni. (Tanuláshoz esetleg alkalmazhatók, de éles környezetben inkább korábbi, stabil változatot használjunk ) 2.1.2. Ingyenes a Drupal? A Drupal azon kívül, hogy sok ember szabadidejébe került, nagyon sok cég fizetett mun. kászinak fizetett munkáját tartalmazza. Pont ez a lényeg, hogy az OpenSource nem egy gáttegyelet, amit próbálnak sulykolni egyes ellenérdekelt cégek, hanem egy üzleti model! (non és for profit) aminek keretében a program forráskódja nem zárt, hanem mindenki számára elérhető. .Az Examiner, Aogua, Lullabot, de akár a magyar cégek, mint pl. az IntegralVision" is je. lentős munkaidőd áldoz a Drupal fejlesztésére a kódok elérhetővé tételére.
2.2. A felhasználó azonosítása 37. oldal 2.2. A felhasználó azonosítása A felhasználó (látogató, 1.5. ábra) )azonosítása azért szükséges, hogy a Drupal el tudja dönteni: mihez van joga a látogatónak. 2.2.1. Regisztráció A Drupal oldalakon a tartalmak beküldése (létrehozása), szerkesztése általában csak regisztrált, és bejelentkezett látogatók számára (vagy azok közül is csak némely szűkebb csoport számára) engedélyezett. (Speciális esetekben a látogatók bejelentkezés nélkül is küldhetnek be tartalmakat: tipikusan fórum bejegyzések, illetve hozzászólások esetén szokás ezt engedélyezni.) A regisztráció során tehát létrejön egy olyan felhasználói fiók, amely a felhasználó szükséges adatait és jogosultságait tartalmazza. A regisztráció – az oldal üzemeltetőjének döntése alapján – háromféle módon történhet: 1. saját magunkat regisztráljuk adminisztrátori elfogadás nélkül 2. saját magunkat regisztráljuk adminisztrátori elfogadással, 3. az adminisztrátor regisztrál. Saját magunkat regisztráljuk adminisztrátori elfogadás nélkül A látogatók maguk végezhetik el a regisztrációt. Ennek módja, hogy a honlap belépésre szolgáló részén megkeressük az Új fiók létrehozása linket (2.1. ábra). 2.1. ábra. Új fiók létrehozása	##### 2.2. A felhasználó azonosítása 37. oldal ## 2.2. A felhasználó azonosítása A felhasználó (látogató, 1.5. ábra) )azonosítása azért szükséges, hogy a Drupal el tudja dönteni: mihez van joga a látogatónak. ### 2.2.1. Regisztráció A Drupal oldalakon a tartalmak beküldése (létrehozása), szerkesztése általában csak regisztrált, és bejelentkezett látogatók számára (vagy azok közül is csak némely szűkebb csoport számára) engedélyezett. (Speciális esetekben a látogatók bejelentkezés nélkül is küldhetnek be tartalmakat: tipikusan fórum bejegyzések, illetve hozzászólások esetén szokás ezt engedélyezni.) A regisztráció során tehát létrejön egy olyan felhasználói fiók, amely a felhasználó szükséges adatait és jogosultságait tartalmazza. A regisztráció – az oldal üzemeltetőjének döntése alapján – háromféle módon történhet: 1. saját magunkat regisztráljuk adminisztrátori elfogadás nélkül 2. saját magunkat regisztráljuk adminisztrátori elfogadással, 3. az adminisztrátor regisztrál. #### Saját magunkat regisztráljuk adminisztrátori elfogadás nélkül A látogatók maguk végezhetik el a regisztrációt. Ennek módja, hogy a honlap belépésre szolgáló részén megkeressük az Új fiók létrehozása linket (2.1. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.2. A felhasználó azonosítása</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">37. oldal</span></p> <p style="top:166.1pt;left:113.6pt;line-height:20.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:20.0pt;color:#000000">2.2. A felhasználó azonosítása</span></b></p> <p style="top:207.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A felhasználó (látogató, 1.5. ábra) )azonosítása azért szükséges, hogy a Drupal el tudja </span></p> <p style="top:219.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">dönteni: mihez van joga a látogatónak.</span></p> <p style="top:247.6pt;left:113.6pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.2.1. Regisztráció</span></b></p> <p style="top:280.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A Drupal oldalakon a tartalmak beküldése (létrehozása), szerkesztése általában csak re-</span></p> <p style="top:292.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">gisztrált, és bejelentkezett látogatók számára (vagy azok közül is csak némely szűkebb cso-</span></p> <p style="top:305.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">port számára) engedélyezett. (Speciális esetekben a látogatók bejelentkezés nélkül is küld-</span></p> <p style="top:317.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">hetnek be tartalmakat: tipikusan fórum bejegyzések, illetve hozzászólások esetén szokás </span></p> <p style="top:330.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ezt engedélyezni.)</span></p> <p style="top:349.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A regisztráció során tehát létrejön egy olyan felhasználói fiók, amely a felhasználó szüksé-</span></p> <p style="top:361.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ges adatait és jogosultságait tartalmazza.</span></p> <p style="top:380.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A regisztráció – az oldal üzemeltetőjének döntése alapján – háromféle módon történhet:</span></p> <p style="top:399.0pt;left:131.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">1.</span></p> <p style="top:399.0pt;left:149.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">saját magunkat regisztráljuk adminisztrátori elfogadás nélkül</span></p> <p style="top:417.8pt;left:131.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">2.</span></p> <p style="top:417.8pt;left:149.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">saját magunkat regisztráljuk adminisztrátori elfogadással,</span></p> <p style="top:436.5pt;left:131.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">3.</span></p> <p style="top:436.5pt;left:149.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">az adminisztrátor regisztrál.</span></p> <p style="top:458.3pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Saját magunkat regisztráljuk adminisztrátori elfogadás nélkül</span></b></p> <p style="top:483.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A látogatók maguk végezhetik el a regisztrációt. Ennek módja, hogy a honlap belépésre </span></p> <p style="top:495.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szolgáló részén megkeressük az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Új fiók létrehozása</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> linket (2.1. ábra).</span></p> <p style="top:702.0pt;left:387.1pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.1. ábra. 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A felhasználó azonosítása 37. oldal 2.2. A felhasználó azonosítása sa azért szükséges, hogy a Drupal el tudja A Drupal oldalakon a tartalmak beküldése (létrehozása), szerkesztése általában csak re- és bejelentkezett látogatók számára (vagy azok közül is csak némely szűkebb cso- port számára) engedélyezett. (Speciális esetekben a látogatók bejelemkezés nélkül is küld- hetnek be tartalmakat: típíkusan fórum bejegyzések, illetve hozzászólások esetén szokás ezt engedélyezni) A regisztráció során tehát létrejön egy olyan felhasználói fiók, amely a felhasználó szüksé-. ges adatait és jogosultságait tartalmazza. A regisztráció - az oldal üzemeltetőjének döntése alapján - háromféle módon történhet: ját magunkat regisztráljuk adminisztrátori elfogadás nélkül magunkat regisztráljuk adminisztrátori elfogadással, 3. az adminisztrátor regisztrál. Saját magunkat regisztráljuk adminisztrátori elfogadás nélkül A látogatók maguk végezhetik el a regisztrációt. Ennek módja, hogy a honlap belépésre szolgáló részén megkeressük az Új fiók létrehozása linket (2.1. ábr 21. ábra. Új fiők létrehozása
38. oldal 2. A Drupal felhasználói szemmel A linkre kattintva megjelenik a Felhasználói fiók oldal (2.2. ábra), ahol a kívánt Felhasználónév és az E-mail cím megadása szükséges. Ezen kívül további adatok megadására is lehet szükség, illetve lehetőség, az adminisztrátor által meghatározott módon. Sajnos egyre gyakrabban van szükség például a Captcha41 ellenőrzés beiktatására, mivel anélkül az egyre intelligensebb spam robotok árasztják el az oldalunkat. A felhasználói név megválasztásánál egyre elterjedtebb megoldás a saját nevünk alkalmazása, főleg olyan oldalaknál, ahol a honlap látogatói nem csak virtuálisan (a honlap látogatóiként), hanem fizikai valójukban is találkozhatnak, ismerhetik egymást. E-mail címként csak a saját, működő e-mail címünket van értelme megadni (2.3. ábra). E lépés célja, hogy korrekt, működő e-mail címmel rendelkezzen minden regisztrált látogató. 41 „A captcha vagy CAPTCHA (magyarosan kapcsa) egy 2000-ben megjelent védekezési módszer a spamek, közelebbről a kommentspamek ellen. A módszer lényege, hogy a hozzászóláshoz a képen látható szót is be kell írni, ez azonban a képfájlon torzítva jelenik meg, tehát a spamrobot nem ismeri föl.” forrás: http://hu.spam.wikia.com/wiki/Captcha 2.2. ábra. Felhasználói fiók létrehozása 2.3. ábra. Felhasználói fiók létrehozása	##### 38. oldal 2. A Drupal felhasználói szemmel A linkre kattintva megjelenik a Felhasználói fiók oldal (2.2. ábra), ahol a kívánt Felhaszná_lónév és az E-mail cím megadása szükséges. Ezen kívül további adatok megadására is lehet_ szükség, illetve lehetőség, az adminisztrátor által meghatározott módon. Sajnos egyre gyakrabban van szükség például a Captcha[41] ellenőrzés beiktatására, mivel anélkül az egyre intelligensebb spam robotok árasztják el az oldalunkat. _2.2. ábra. Felhasználói fiók létrehozása_ A felhasználói név megválasztásánál egyre elterjedtebb megoldás a saját nevünk alkalmazása, főleg olyan oldalaknál, ahol a honlap látogatói nem csak virtuálisan (a honlap látogatóiként), hanem fizikai valójukban is találkozhatnak, ismerhetik egymást. E-mail címként csak a saját, működő e-mail címünket van értelme megadni (2.3. ábra). E lépés célja, hogy korrekt, működő e-mail címmel rendelkezzen minden regisztrált látogató. _2.3. ábra. Felhasználói fiók létrehozása_ 41 „A captcha vagy CAPTCHA (magyarosan kapcsa) egy 2000-ben megjelent védekezési módszer a spamek, _közelebbről a kommentspamek ellen. A módszer lényege, hogy a hozzászóláshoz a képen látható szót is be_ _kell írni, ez azonban a képfájlon torzítva jelenik meg, tehát a spamrobot nem ismeri föl.” forrás:_ http://hu.spam.wikia.com/wiki/Captcha -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">38. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:147.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A linkre kattintva megjelenik a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Felhasználói fiók</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> oldal (2.2. ábra), ahol a kívánt </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Felhaszná-</span></i></p> <p style="top:159.9pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">lónév</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> és az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">E-mail cím</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> megadása szükséges. Ezen kívül további adatok megadására is lehet </span></p> <p style="top:172.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szükség, illetve lehetőség, az adminisztrátor által meghatározott módon. Sajnos egyre </span></p> <p style="top:185.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">gyakrabban van szükség például a Captcha</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">41</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> ellenőrzés beiktatására, mivel anélkül az egy-</span></p> <p style="top:197.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">re intelligensebb spam robotok árasztják el az oldalunkat.</span></p> <p style="top:420.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A felhasználói név megválasztásánál egyre elterjedtebb megoldás a saját nevünk alkalma-</span></p> <p style="top:432.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">zása, főleg olyan oldalaknál, ahol a honlap látogatói nem csak virtuálisan (a honlap látoga-</span></p> <p style="top:445.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tóiként), hanem fizikai valójukban is találkozhatnak, ismerhetik egymást.</span></p> <p style="top:464.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">E-mail címként csak a saját, működő e-mail címünket van értelme megadni (2.3. ábra). E </span></p> <p style="top:476.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lépés célja, hogy korrekt, működő e-mail címmel rendelkezzen minden regisztrált látogató.</span></p> <p style="top:688.4pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">41</span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:9.0pt;color:#000000"> „A captcha vagy CAPTCHA (magyarosan kapcsa) egy 2000-ben megjelent védekezési módszer a spamek, </span></i></p> <p style="top:698.6pt;left:99.3pt;line-height:9.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:9.0pt;color:#000000">közelebbről a kommentspamek ellen. A módszer lényege, hogy a hozzászóláshoz a képen látható szót is be </span></i></p> <p style="top:708.9pt;left:99.3pt;line-height:9.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:9.0pt;color:#000000">kell írni, ez azonban a képfájlon torzítva jelenik meg, tehát a spamrobot nem ismeri föl.”</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> forrás: </span></p> <p style="top:719.9pt;left:99.3pt;line-height:8.0pt"><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c">http://hu.spam.wikia.com/wiki/Captcha</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:396.3pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.2. ábra. 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A Drupal felhasználói szememel A linkre kattintva megjelenik a Felkasználói fiók oldal (2:2. ábra), ahol a kívánt Felhaszná- Tőnév és az E-mail cím megadása szükséges. Ezen kívül további adatok megadására is lehet szükség, illetve lehetőség, az adminisztrátor által meghatározott módon. Sajnos egyre. agyakrabban van szűkség például a Captha" ellenőrzés beíktatására, mivel anélki Te intelligensebb spam robotok árasztják el az oldalunkat. Felhasználói fiók jtzezss né újezssen azzs— 22 ábra. Felhasználói fiők létrehozása A felhasználói név megválasztásánál egyre elterjedtebb megoldás a saját nevünk alkalma- zása, föleg alyan oldalaknál, ahol a honlap látogatói nem csak virtválisan (a honlap látoga. tóikénd), hanem fizikai valójukban is találkozhatnak, ismerhetik egymást. ímként csak a saját, működő e-mail címűnket van értelme megadni (2.3. ábra). E. lépés célja hogy korrekt, működő e-mail címmel vendelkezzen minden regiszttált látogató. Felhasználói fiók a— 23. ábra. Felhasználói fiők lérrehozása -3. A czptcha vagy CAPTCHA (rmagyarosan kapcsa) egy 2000-ben megjelent védekezési módszer a spamek, Fözelebbrál a kormentspamek elen A mádszer lényege, hogy a hozzószításhoz a képen tíható szót is be kell ír cz azonban a képfájlon torítoa jelendk meg tehát a spamrobas nem ter öl" forrás höpívoszamalkasomhnkíszs
42. oldal 2. A Drupal felhasználói szemmel Az adminisztrátor regisztrál Előfordulhat, hogy az adminisztrátor maga hoz létre a felhasználók számára felhasználói azonosítót. Ebben az esetben a Drupal (vagy az adminisztrátor) egy e-mailben értesíti (2.12. ábra) a leendő felhasználót a regisztráció megtörténtéről. Ennek előnye az is, hogy a felhasználó megfelelő jogosultságait már ekkor megkaphatja. Zárt oldalakra is többnyire így lehet bekerülni. Az OpenID használata Technikailag létező, de Magyarországon alig ismert megoldás az OpenID használata. A Wikipédia42 szerint „az OpenID egy nyílt, decentralizált, ingyenes internetes szolgáltatás, ami lehetővé teszi a felhasználók számára, hogy egyetlen digitális identitással lépjenek be 42 http://hu.wikipedia.org/wiki/OpenID 2.11. ábra. Jelszó első megadása 2.12. ábra. Adminisztrátor által létrehozott felhasználói fiók	##### 42. oldal 2. A Drupal felhasználói szemmel _2.11. ábra. Jelszó első megadása_ #### Az adminisztrátor regisztrál Előfordulhat, hogy az adminisztrátor maga hoz létre a felhasználók számára felhasználói azonosítót. Ebben az esetben a Drupal (vagy az adminisztrátor) egy e-mailben értesíti (2.12. ábra) a leendő felhasználót a regisztráció megtörténtéről. Ennek előnye az is, hogy a felhasználó megfelelő jogosultságait már ekkor megkaphatja. Zárt oldalakra is többnyire így lehet bekerülni. _2.12. ábra. Adminisztrátor által létrehozott felhasználói fiók_ #### Az OpenID használata Technikailag létező, de Magyarországon alig ismert megoldás az OpenID használata. A Wikipédia[42] szerint „az OpenID egy nyílt, decentralizált, ingyenes internetes szolgáltatás, _ami lehetővé teszi a felhasználók számára, hogy egyetlen digitális identitással lépjenek be_ 42 http://hu.wikipedia.org/wiki/OpenID -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">42. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:347.7pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Az adminisztrátor regisztrál</span></b></p> <p style="top:372.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Előfordulhat, hogy az adminisztrátor maga hoz létre a felhasználók számára felhasználói </span></p> <p style="top:385.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">azonosítót. Ebben az esetben a Drupal (vagy az adminisztrátor) egy e-mailben értesíti </span></p> <p style="top:397.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(2.12. ábra) a leendő felhasználót a regisztráció megtörténtéről. Ennek előnye az is, hogy a </span></p> <p style="top:410.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">felhasználó megfelelő jogosultságait már ekkor megkaphatja. Zárt oldalakra is többnyire </span></p> <p style="top:422.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">így lehet bekerülni.</span></p> <p style="top:638.3pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Az OpenID használata</span></b></p> <p style="top:663.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Technikailag létező, de Magyarországon alig ismert megoldás az OpenID használata. A </span></p> <p style="top:675.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Wikipédia</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">42</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szerint </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">„az OpenID egy nyílt, decentralizált, ingyenes internetes szolgáltatás, </span></i></p> <p style="top:688.1pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">ami lehetővé teszi a felhasználók számára, hogy egyetlen digitális identitással lépjenek be </span></i></p> <p style="top:719.1pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">42</span><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c"> http://hu.wikipedia.org/wiki/OpenID</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:323.6pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.11. ábra. 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A Drupal felhasználói szememel "Teszt Elek2 D— .211. ábra. Jelszó első megadása Az adminisztrátor regisztrál Előfordulhat, hogy az adminisztrátor maga hoz létre a felhasználók számára felhasználói azonosítót, Ebben az esetben a Drupal (vagy az adminisztrátor) egy e-mallben értesi 212. ábra) a leendő felhasználót a regisztráció megtörténtéről. Ennek előnye az is, hogy a felhasználó megfelelő jogosultságait már ekkor megkaphatja. Zárt oldalakra is többnyire. így lehet bekerülni. ] Teszt Elka felhasználót rehezta a pelhely sámiriszbálora A bozpszhez az aabb Mztsozasr kel kazstani vagy b kl másokó sbongészó cmserába Eztoratkozás csak ogyszot használató belépósra a betotom olőalon meghaő sdr a jelzők Abésőltiekben a hörelonát AzOAPTLT ÁSE cn tözénő kejelenősezéshez az l megséct klzőt tel hasznáti. Folhasználánér TesztElekő ökzó a megadot plszó — Dnupol T tesztcsapota 232 ábra. Adminisztrátor által létrehozott felhasználói fiők. Az OpenlD használata Technikailag létező, de Magyarországon alig ismert megoldás az OpenlD használata. A MWikipédia" szerint , az OpenlD egy nyílt, decentralizált, ingyenes intemetes szolgáltatás, aami lehetővé teszi a felhasználók számára, hogy egyetlen digítális identítással lépjenek be eee ]
2.2. A felhasználó azonosítása 43. oldal különböző oldalakra”. Természetesen a Drupal alkalmas az OpenID bejelentkezések kezelésére. A 2.13. ábrán látható módon látszik, ha ez a szolgáltatás elérhető a weboldalon. 2.2.2. Be- és kijelentkezés Addig, amíg az oldalra be nem jelentkezünk a felhasználónév és jelszó megadásával, mindössze azonosítatlan (anonymous, a továbbiakban névtelen vagy vendég) felhasználóként tudjuk az oldalt használni. Ha ki akarjuk használni a regisztrált felhasználói azonosítónkkal járó plusz szolgáltatásokat, akkor mindenképpen be kell jelentkeznünk. A bejelentkezés legegyszerűbb módja, hogy a 2.1. ábrán látható űrlapon megadjuk a felhasználónevünket és a jelszavunkat, majd a Bejelentkezés gombra kattintunk. A sikeres belépésre utal többek között, hogy az eddig látható Bejelentkezés űrlap (célja nem lévén) nem lesz látható. Látszik viszont helyette a Saját adatok és a Kilépés menüpont (2.14. ábra). 2.13. ábra. OpenID 2.14. ábra. Saját adatok és Kilépés menüpontok	##### 2.2. A felhasználó azonosítása 43. oldal _különböző oldalakra”. Természetesen a Drupal alkalmas az OpenID bejelentkezések kezelé-_ sére. A 2.13. ábrán látható módon látszik, ha ez a szolgáltatás elérhető a weboldalon. _2.13. ábra. OpenID_ ### 2.2.2. Be- és kijelentkezés Addig, amíg az oldalra be nem jelentkezünk a felhasználónév és jelszó megadásával, mindössze azonosítatlan (anonymous, a továbbiakban névtelen vagy vendég) felhasználóként tudjuk az oldalt használni. Ha ki akarjuk használni a regisztrált felhasználói azonosítónkkal járó plusz szolgáltatásokat, akkor mindenképpen be kell jelentkeznünk. A bejelentkezés legegyszerűbb módja, hogy a 2.1. ábrán látható űrlapon megadjuk a felhasználónevünket és a jelszavunkat, majd a Bejelentkezés gombra kattintunk. A sikeres belépésre utal többek között, hogy az eddig látható Bejelentkezés űrlap (célja nem lévén) nem lesz látható. Látszik viszont helyette a Saját adatok és a Kilépés menüpont (2.14. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.2. A felhasználó azonosítása</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">43. oldal</span></p> <p style="top:147.4pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">különböző oldalakra”</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">. Természetesen a Drupal alkalmas az OpenID bejelentkezések kezelé-</span></p> <p style="top:159.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">sére. A 2.13. ábrán látható módon látszik, ha ez a szolgáltatás elérhető a weboldalon.</span></p> <p style="top:420.5pt;left:113.6pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.2.2. Be- és kijelentkezés</span></b></p> <p style="top:452.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Addig, amíg az oldalra be nem jelentkezünk a felhasználónév és jelszó megadásával, mind-</span></p> <p style="top:465.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">össze azonosítatlan (anonymous, a továbbiakban névtelen vagy vendég) felhasználóként </span></p> <p style="top:478.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tudjuk az oldalt használni. Ha ki akarjuk használni a regisztrált felhasználói azonosítónk-</span></p> <p style="top:490.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kal járó plusz szolgáltatásokat, akkor mindenképpen be kell jelentkeznünk.</span></p> <p style="top:509.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A bejelentkezés legegyszerűbb módja, hogy a 2.1. ábrán látható űrlapon megadjuk a fel-</span></p> <p style="top:521.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">használónevünket és a jelszavunkat, majd a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Bejelentkezés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombra kattintunk.</span></p> <p style="top:540.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A sikeres belépésre utal többek között, hogy az eddig látható </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Bejelentkezés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> űrlap (célja </span></p> <p style="top:553.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nem lévén) nem lesz látható. Látszik viszont helyette a</span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000"> Saját adatok</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> és a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Kilépés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> menüpont </span></p> <p style="top:565.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(2.14. ábra).</span></p> <p style="top:396.1pt;left:386.5pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.13. ábra. 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A felhasználó azonosítása 43. oldal különböző oldalakra". Természetesen a Drupal alkalmas az OpenlD bejelentkezések kezelé- sére. A 2.13. ábrán látható módon látszik, ha ez a szolgáltatás elérhető a weboldalon. 213. ábra. OpenlD Be- és kijelentkezés Addig, amig az oldalra be nem jelentkezünk a felhasználónév és jelszó megadásával, mind- össze azonosítalan (anonymous, a továbbiakban névtelen vagy vendég) felhasználóként tudjuk az oldalt használni. Ha ki akarjuk használni a regisztrált felhasználói azonosítónk- kal járó plusz szolgáltatásokat, akkor mindenképpen be kell jelentkeznünk. A bejelentkezés legegyszerűbb módja, hogy a 2.1. ábrán látható űrlapon megadjuk a fel- használónevünket és a jelszavunkat, majd a Bejelentkezés gombra kattintunk. A sikeres belépésre utal többek között, hogy az eddig látható Bejelentkezés űrlap (célja mem lévén) nem lesz látható. Látszik viszont helyette a Saját adatok és a Kilépés menüpont (214. ábra). p , . Drupal T teszt Üdvözlet Drupal 7 teszt webhelyen. 214. ábra. Szját adatok és Kilépés menüpontok
44. oldal 2. A Drupal felhasználói szemmel A Kilépés menüpontra kattintva ismét névtelen felhasználóvá válunk a Drupal alapú oldal számára. A böngészőnk (beállításaitól függően) felajánlhatja, hogy a begépelt adatokat megjegyzi. Ezt csak akkor fogadjuk el, ha a számítógéphez fizikailag más nem tud hozzáférni. Például internetkávézóban, iskolai gépteremben nem szabad engedélyeznünk, mert akkor illetéktelenek használhatják a honlapot a mi nevünkben és jogosultságunkkal. Ha engedélyezzük a belépési adatok megjegyzését, akkor a legközelebbi látogatáskor a böngészőnk fel fogja ajánlani a korábbi adatokat, így azokat nem kell újra begépelnünk. Biztonsági okokból lehetőleg mindig lépjünk ki a Kilépés link (2.14. ábra) segítségével. 2.2.3. Saját adatok módosítása A regisztrált felhasználók saját adataikat megváltoztathatják a Saját adatok (2.14. ábra) linkre, majd a Szerkesztés fülre kattintva. Az e-mail cím és a jelszó megváltoztatása minden esetben lehetséges. Jelszó változtatása esetén ismét meg kell adnunk a régi jelszavunkat is (2.16. ábra). 2.15. ábra. Jelszó megjegyzése	##### 44. oldal 2. A Drupal felhasználói szemmel A Kilépés menüpontra kattintva ismét névtelen felhasználóvá válunk a Drupal alapú oldal számára. A böngészőnk (beállításaitól függően) felajánlhatja, hogy a begépelt adatokat megjegyzi. Ezt csak akkor fogadjuk el, ha a számítógéphez fizikailag más nem tud hozzáférni. Például internetkávézóban, iskolai gépteremben nem szabad engedélyeznünk, mert akkor illetéktelenek használhatják a honlapot a mi nevünkben és jogosultságunkkal. _2.15. ábra. Jelszó megjegyzése_ Ha engedélyezzük a belépési adatok megjegyzését, akkor a legközelebbi látogatáskor a böngészőnk fel fogja ajánlani a korábbi adatokat, így azokat nem kell újra begépelnünk. Biztonsági okokból lehetőleg mindig lépjünk ki a Kilépés link (2.14. ábra) segítségével. ### 2.2.3. Saját adatok módosítása A regisztrált felhasználók saját adataikat megváltoztathatják a Saját adatok (2.14. ábra) linkre, majd a Szerkesztés fülre kattintva. Az e-mail cím és a jelszó megváltoztatása minden esetben lehetséges. Jelszó változtatása esetén ismét meg kell adnunk a régi jelszavunkat is (2.16. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">44. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:147.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Kilépés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> menüpontra kattintva ismét névtelen felhasználóvá válunk a Drupal alapú oldal </span></p> <p style="top:159.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">számára.</span></p> <p style="top:178.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A böngészőnk (beállításaitól függően) felajánlhatja, hogy a begépelt adatokat megjegyzi. </span></p> <p style="top:191.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ezt csak akkor fogadjuk el, ha a számítógéphez fizikailag más nem tud hozzáférni. Például </span></p> <p style="top:203.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">internetkávézóban, iskolai gépteremben nem szabad engedélyeznünk, mert akkor illetékte-</span></p> <p style="top:216.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lenek használhatják a honlapot a mi nevünkben és jogosultságunkkal.</span></p> <p style="top:470.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha engedélyezzük a belépési adatok megjegyzését, akkor a legközelebbi látogatáskor a </span></p> <p style="top:482.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">böngészőnk fel fogja ajánlani a korábbi adatokat, így azokat nem kell újra begépelnünk.</span></p> <p style="top:501.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Biztonsági okokból lehetőleg mindig lépjünk ki a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Kilépés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> link (2.14. ábra) segítségével.</span></p> <p style="top:529.5pt;left:85.2pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.2.3. Saját adatok módosítása</span></b></p> <p style="top:561.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A regisztrált felhasználók saját adataikat megváltoztathatják a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Saját adatok</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> (2.14. ábra) </span></p> <p style="top:574.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">linkre, majd a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Szerkesztés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> fülre kattintva.</span></p> <p style="top:593.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az e-mail cím és a jelszó megváltoztatása minden esetben lehetséges. Jelszó változtatása </span></p> <p style="top:605.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">esetén ismét meg kell adnunk a régi jelszavunkat is (2.16. ábra).</span></p> <p style="top:446.3pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.15. ábra. 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A Drupal felhasználói szemenel A Kilépés menüpontra kattintva ismét névtelen felhasználóvá válunk a Drupal alapú oldal A böngészőnk (beállításaitól függően) felajánlhatja, hogy a begépelt adatokat megjegyzi. Ezt csak akkor fogadjuk el, ha a számítógéphez fizikailag más nem tud hozzáférni. Például internetkávézőban, iskolai gépteremben nem szabad engedélyeznünk, mert akkor illetékte- lenek használhatják a honlapot a mi nevünkben és jogosultságunkkal 215. ábra. Jelszó megjegyzése Ha engedélyezzük a belépési adatok megjegyzését, akkor a legközelebbi látogatáskor a böngészőnk fel fogja ajánlani a kozábbi adatokat, így azokat nem kell újra begépelnünk. Biztonsági okokból lehetőleg mindig lépjünk ki a Kilépés ink (2.14. ábbra) segítségével. Saját adatok módosítása A regásztrált felhasználók saját adataikat megváltoztathatják a Sajár adatok (2.14. ábra) linkre, majd a Szerkesztés fülre kattintva. .Az e-mail cím és a jelszó megváltoztatása minden esetben lehetséges. Jelszó változtatása esetén ismét meg kell adnunk a régi jelszavunkat is (2-16. ábra).
2.2. A felhasználó azonosítása 45. oldal A jelszó kiválasztásánál érdemes az erősségre is figyelni. Ötleteket is kaphatunk a komplexitás növelésére. Az adminisztrátor beállításaitól függ, hogy pontosan ezen kívül mit tudunk az oldalon beállítani. A következők szoktak előfordulni (2.17. ábra): Ha engedélyezve van, megváltoztathatjuk a felhasználónevünket43. Ha engedélyezve van, itt feltölthetünk egy saját arcképet, ami például a beküldött tartalmaink, hozzászólásaink mellett jelenhet meg. Többnyelvű oldal esetén a felhasználói felület nyelvét megváltoztathatjuk. Ha engedélyezve van, az időzóna megadásával korrigálhatjuk a szerver és a mi szá- mítógépünk közötti esetleges időzóna-eltérést. Ha az oldal többféle kinézettel (sminkkel) rendelkezik, beállíthatjuk a számunkra megfelelőt. Ha engedélyezve van, a hozzászólásainknál megjelenő aláírás szöveget is megadha- tunk. 43 Ezt ritkán szoktunk engedélyezni, inkább az adminisztrátor hatáskörében hagyjuk ezt a jogot. 2.16. ábra. Saját adatok szerkesztése	##### 2.2. A felhasználó azonosítása 45. oldal _2.16. ábra. Saját adatok szerkesztése_ A jelszó kiválasztásánál érdemes az erősségre is figyelni. Ötleteket is kaphatunk a komplexitás növelésére. Az adminisztrátor beállításaitól függ, hogy pontosan ezen kívül mit tudunk az oldalon beállítani. A következők szoktak előfordulni (2.17. ábra):  Ha engedélyezve van, megváltoztathatjuk a felhasználónevünket[43].  Ha engedélyezve van, itt feltölthetünk egy saját arcképet, ami például a beküldött tartalmaink, hozzászólásaink mellett jelenhet meg.  Többnyelvű oldal esetén a felhasználói felület nyelvét megváltoztathatjuk.  Ha engedélyezve van, az időzóna megadásával korrigálhatjuk a szerver és a mi szá mítógépünk közötti esetleges időzóna-eltérést.  Ha az oldal többféle kinézettel (sminkkel) rendelkezik, beállíthatjuk a számunkra megfelelőt.  Ha engedélyezve van, a hozzászólásainknál megjelenő aláírás szöveget is megadha -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.2. A felhasználó azonosítása</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">45. oldal</span></p> <p style="top:444.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A jelszó kiválasztásánál érdemes az erősségre is figyelni. Ötleteket is kaphatunk a komple-</span></p> <p style="top:457.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">xitás növelésére.</span></p> <p style="top:476.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az adminisztrátor beállításaitól függ, hogy pontosan ezen kívül mit tudunk az oldalon be-</span></p> <p style="top:488.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">állítani. A következők szoktak előfordulni (2.17. ábra):</span></p> <p style="top:508.9pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha engedélyezve van, megváltoztathatjuk a felhasználónevünket</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">43</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">.</span></p> <p style="top:529.2pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha engedélyezve van, itt feltölthetünk egy saját arcképet, ami például a beküldött </span></p> <p style="top:541.7pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tartalmaink, hozzászólásaink mellett jelenhet meg.</span></p> <p style="top:562.0pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Többnyelvű oldal esetén a felhasználói felület nyelvét megváltoztathatjuk.</span></p> <p style="top:582.2pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha engedélyezve van, az időzóna megadásával korrigálhatjuk a szerver és a mi szá-</span></p> <p style="top:594.8pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">mítógépünk közötti esetleges időzóna-eltérést.</span></p> <p style="top:615.0pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha az oldal többféle kinézettel (sminkkel) rendelkezik, beállíthatjuk a számunkra </span></p> <p style="top:627.6pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">megfelelőt.</span></p> <p style="top:647.8pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha engedélyezve van, a hozzászólásainknál megjelenő aláírás szöveget is megadha-</span></p> <p style="top:660.4pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tunk.</span></p> <p style="top:719.1pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">43 Ezt ritkán szoktunk engedélyezni, inkább az adminisztrátor hatáskörében hagyjuk ezt a jogot.</span></p> <p style="top:420.9pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.16. ábra. 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A felhasználó azonosítása 45. oldal 216. ábra. Saját adatok szerkesztése A jelszó kiválasztásánál érdemes az erősségre is figyelni. Ötleteket is kaphatunk a komple- xitás növelésére. .Az adminisztrátor beállításaitól függ, hogy pontosan ezen kívül mit tudunk az oldalon be- állítani. A következők szoktak előfordulni (2.17. ábra) Ha engedélyezve van, megváltoztathatjuk a felhasználónevűnket". Ha engedélyezve van, ítt feltölthetünk egy saját areképet, ami például a beküldött. tartalmaink, hozzászólásaink mellett jelenhet meg, "Többnyelvű oldal esetén a felhasználói felület nyelvét megváltoztathatjuk. Ha engedélyezve van, az időzóna megadásával korrigálhatjuk a szerver és a mi szá- mitógépünk közötti esetleges időzóna-eltérést. Ha az oldal többféle kinézettel (sminkkel) rendelkezik, beállíthat megfelelőt. ik a számunkra Ha engedélyezve van, a hozzászólásainknál megjelenő aláírás szöveget is megadhi tunk 3 Fa valán szoktuak cagodélyezni, inkább az nárieisztrátot hatáskörében hagyjok ezt a jogot.
2.2. A felhasználó azonosítása 47. oldal Új jelszó igénylése Egyszerűbb esetben a 2.1. ábrán látható módon elérhetjük ezt a funkciót. Ha esetleg ez a belépés blokk nem látszik az oldalon, az user útvonallal is próbálkozhatunk: a böngésző cím sorába írjuk be a domain név után az user útvonalat. (A szerző honlapján pl. http://nagygusztav.hu/user lesz.) Az itt megjelenő űrlapot láttuk már a 2.2. ábrán. Ott is jól látszik az Új jelszó igénylése fül, amelyre kattintva megadhatjuk az e-mail címünket vagy felhasználói nevünket (2.18. ábra). A szöveg sikeres begépelése után egy rövid üzenetet kapunk: „A további teendők leírása nemsokára e-mailben érkezik.” Ennek megfelelően egy e-mailt fogunk kapni (2.19. ábra). Ha egy napon belül nem kattintunk az e-mailben kapott belépési linkre, akkor semmi következménye nem lesz az e-mail kérésének, a korábbi felhasználónévvel és jelszóval be tudunk jelentkezni. Ha kattintunk, a 2.20. ábrához hasonló üzenetet kapunk. Természetesen jelentkezzünk be, majd a 2.16. ábrához hasonló módon állítsunk be egy új jelszót. A továbbiakban ezzel fogunk tudni belépni. 2.18. ábra. Új jelszó igénylése 2.19. ábra. E-mail az új jelszó igénylése esetén	##### 2.2. A felhasználó azonosítása 47. oldal #### Új jelszó igénylése Egyszerűbb esetben a 2.1. ábrán látható módon elérhetjük ezt a funkciót. Ha esetleg ez a belépés blokk nem látszik az oldalon, az user útvonallal is próbálkozhatunk: a böngésző cím sorába írjuk be a domain név után az user útvonalat. (A szerző honlapján pl. http://nagygusztav.hu/user lesz.) Az itt megjelenő űrlapot láttuk már a 2.2. ábrán. Ott is jól látszik az Új jelszó igénylése fül, amelyre kattintva megadhatjuk az e-mail címünket vagy felhasználói nevünket (2.18. ábra). _2.18. ábra. Új jelszó igénylése_ A szöveg sikeres begépelése után egy rövid üzenetet kapunk: „A további teendők leírása nemsokára e-mailben érkezik.” Ennek megfelelően egy e-mailt fogunk kapni (2.19. ábra). _2.19. ábra. E-mail az új jelszó igénylése esetén_ Ha egy napon belül nem kattintunk az e-mailben kapott belépési linkre, akkor semmi következménye nem lesz az e-mail kérésének, a korábbi felhasználónévvel és jelszóval be tudunk jelentkezni. Ha kattintunk, a 2.20. ábrához hasonló üzenetet kapunk. Természetesen jelentkezzünk be, majd a 2.16. ábrához hasonló módon állítsunk be egy új jelszót. A továbbiakban ezzel fogunk tudni belépni. -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.2. A felhasználó azonosítása</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">47. oldal</span></p> <p style="top:153.5pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Új jelszó igénylése</span></b></p> <p style="top:178.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Egyszerűbb esetben a 2.1. ábrán látható módon elérhetjük ezt a funkciót. Ha esetleg ez a </span></p> <p style="top:190.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">belépés blokk nem látszik az oldalon, az </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">user</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> útvonallal is próbálkozhatunk: a böngésző </span></p> <p style="top:203.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">cím sorába írjuk be a domain név után az </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">user</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> útvonalat. (A szerző honlapján pl. </span></p> <p style="top:216.7pt;left:113.6pt;line-height:10.0pt"><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">http://nagygusztav.hu/user</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> lesz.) Az itt megjelenő űrlapot láttuk már a 2.2. ábrán. Ott is jól </span></p> <p style="top:228.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">látszik az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Új jelszó igénylése</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> fül, amelyre kattintva megadhatjuk az e-mail címünket vagy </span></p> <p style="top:241.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">felhasználói nevünket (2.18. ábra).</span></p> <p style="top:409.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A szöveg sikeres begépelése után egy rövid üzenetet kapunk: „A további teendők leírása </span></p> <p style="top:422.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nemsokára e-mailben érkezik.” Ennek megfelelően egy e-mailt fogunk kapni (2.19. ábra).</span></p> <p style="top:612.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha egy napon belül nem kattintunk az e-mailben kapott belépési linkre, akkor semmi kö-</span></p> <p style="top:625.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">vetkezménye nem lesz az e-mail kérésének, a korábbi felhasználónévvel és jelszóval be tu-</span></p> <p style="top:637.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">dunk jelentkezni.</span></p> <p style="top:656.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha kattintunk, a 2.20. ábrához hasonló üzenetet kapunk.</span></p> <p style="top:675.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Természetesen jelentkezzünk be, majd a 2.16. ábrához hasonló módon állítsunk be egy új </span></p> <p style="top:687.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">jelszót. A továbbiakban ezzel fogunk tudni belépni.</span></p> <p style="top:385.7pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.18. ábra. 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A felhasználó azonosítása 47. oldal ÚJ jelszó igénylése Egyszerűbb esetben a 2.. ábrán látható módon elérhetjük ezt a funkciót. Ha esetleg ez a belépés blokk nem látszik az oldalon, az user útvonallal is próbálkozhatunk: a bőngésző cím sorába írjuk be a domain név után az user útvonalat, (A szerző honlapján pl. btp//9agygusztavhufuser lesz) Az itt megjelenő űrlapot láttuk már a 2.2. ábrán. Ott is jól "Felhasználói fiók 218. ábra. Új jelsző igénylése A szöveg sikeres begépelése után egy rövid üzenetet kapunk: A további teendők leírása memsokára e-mailben érkezik" Ennek megfelelően egy e-mailt fogunk kapni (2-19. ábra). — sagyosszan ezösn sothlyot emnes a fasznknek né a Joszó vart gény t boryata ] F ] zegy egyszs hasznünató boéés més £ rap l érvényessége lr 05 sers er TÖnénL 1a né haszoáták t A öndyén a mzéfel bb nn or t s sönlk aZ. 1001 jok vog ot n vlnszai a hazesmészancsom 219. ábra. E-mail az új jelszó igénylése esetén. Ha egy napon belül nem kattintunk az e-mailben kapott bel vetkezménye nem lesz az e-mail kérésének, a korábbi felhasználónévvel és jelszóval be tu- dunk jelentkezni. Ha kattintunk, a 220. ábrához hasonló üzenetet kapunk. ihoz hasonló módon állítsunk e egy új Természetesen jelentkezzünk be, majd a 2.16. ál jelszót. A továbbiakban ezzel fogunk tudni belépni.
48. oldal 2. A Drupal felhasználói szemmel 2.3. Tartalmak kezelése A Drupal tartalomkezelő rendszer fő célja, hogy a honlap tartalmait (oldalait) kezelje, vagyis lehetővé tegye az oldalak létrehozását, módosítását, törlését, megtekintését. (Természetesen a szolgáltatásokat csak az adott feladat ellátására jogosult felhasználók érhetik el.) 2.3.1. Tartalmak megtekintése Ez az a funkció, amit minden webet használó jól ismer. Az egyes oldalak tartalmait többféle módon érhetjük el. Tipikus lehetőségek: A címlapon találhatóak általában a legfrissebb tartalmak címei, bevezetői. (Blogok esetén nem csak bevezető, hanem a teljes tartalom megjelenítése is szokásos.) Jellemzően felül és bal (esetleg jobb) oldalt megjelenő menüpontok közvetlenül vagy közvetve újabb oldalakra vezethetnek. Egyre jellemzőbb, hogy többféle navigációs eszközzel is elérhetünk egyes tartalma- kat, pl. címkék, címkefelhő, kenyérmorzsa menü, stb. 2.3.2. Tartalmak létrehozása Amennyiben rendelkezünk megfelelő jogosultságokkal, a Navigáció menün megjelenik a Tartalom hozzáadása link (2.21. ábra). 2.20. ábra. Az e-mailben kapott linkre kattintva	##### 48. oldal 2. A Drupal felhasználói szemmel _2.20. ábra. Az e-mailben kapott linkre kattintva_ ## 2.3. Tartalmak kezelése A Drupal tartalomkezelő rendszer fő célja, hogy a honlap tartalmait (oldalait) kezelje, vagyis lehetővé tegye az oldalak létrehozását, módosítását, törlését, megtekintését. (Természetesen a szolgáltatásokat csak az adott feladat ellátására jogosult felhasználók érhetik el.) ### 2.3.1. Tartalmak megtekintése Ez az a funkció, amit minden webet használó jól ismer. Az egyes oldalak tartalmait többféle módon érhetjük el. Tipikus lehetőségek:  A címlapon találhatóak általában a legfrissebb tartalmak címei, bevezetői. (Blogok esetén nem csak bevezető, hanem a teljes tartalom megjelenítése is szokásos.)  Jellemzően felül és bal (esetleg jobb) oldalt megjelenő menüpontok közvetlenül vagy közvetve újabb oldalakra vezethetnek.  Egyre jellemzőbb, hogy többféle navigációs eszközzel is elérhetünk egyes tartalma kat, pl. címkék, címkefelhő, kenyérmorzsa menü, stb. ### 2.3.2. Tartalmak létrehozása Amennyiben rendelkezünk megfelelő jogosultságokkal, a Navigáció menün megjelenik a _Tartalom hozzáadása link (2.21. ábra)._ -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">48. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:342.8pt;left:85.2pt;line-height:20.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:20.0pt;color:#000000">2.3. Tartalmak kezelése</span></b></p> <p style="top:383.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A Drupal tartalomkezelő rendszer fő célja, hogy a honlap tartalmait (oldalait) kezelje, </span></p> <p style="top:396.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">vagyis lehetővé tegye az oldalak létrehozását, módosítását, törlését, megtekintését. (Termé-</span></p> <p style="top:408.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szetesen a szolgáltatásokat csak az adott feladat ellátására jogosult felhasználók érhetik el.)</span></p> <p style="top:436.8pt;left:85.2pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.3.1. Tartalmak megtekintése</span></b></p> <p style="top:469.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ez az a funkció, amit minden webet használó jól ismer. Az egyes oldalak tartalmait többfé-</span></p> <p style="top:481.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">le módon érhetjük el. Tipikus lehetőségek:</span></p> <p style="top:502.1pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A címlapon találhatóak általában a legfrissebb tartalmak címei, bevezetői. (Blogok </span></p> <p style="top:514.6pt;left:113.5pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">esetén nem csak bevezető, hanem a teljes tartalom megjelenítése is szokásos.)</span></p> <p style="top:534.9pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Jellemzően felül és bal (esetleg jobb) oldalt megjelenő menüpontok közvetlenül vagy </span></p> <p style="top:547.4pt;left:113.5pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">közvetve újabb oldalakra vezethetnek.</span></p> <p style="top:567.7pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Egyre jellemzőbb, hogy többféle navigációs eszközzel is elérhetünk egyes tartalma-</span></p> <p style="top:580.2pt;left:113.5pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kat, pl. címkék, címkefelhő, kenyérmorzsa menü, stb.</span></p> <p style="top:608.2pt;left:85.2pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.3.2. Tartalmak létrehozása</span></b></p> <p style="top:640.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Amennyiben rendelkezünk megfelelő jogosultságokkal, a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Navigáció</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> menün megjelenik a </span></p> <p style="top:653.2pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Tartalom hozzáadása</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> link (2.21. ábra).</span></p> <p style="top:318.0pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.20. ábra. 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A Drupal felhasználói szemmel Jelszó átállítása legyegyszetbaszzálbató belépés mód Nagy Gasztás részére, és 2091 9907. 11 időpoztban kefog Azalábbs orebrskarztsalebet a metbelre bejlsntkezni és a jeszót megyáltoztati a belépési mód csak egyzethasználbató. erentezés 220. ábra. Az e-mailben kapott lnkre kattintva 2.3. Tartalmak kezelése A Drupal tartalomkezelő rendszer fő célja, hogy a honlap tartalmait (oldalait) kezelje, vagyis lehetővé tes ú, törlését, megtekintését. (Termé. szetesen a szolgált: jogosult felhasználók érhetik el.) 2.3.1. Tartalmak megtekintése Ez az a funkció, amit minden wwebet használó jól ismer. Az egyes oldalak tartalmait többfé. le módon érketjük el. Tipikus lehetőségek; m A címlapon találhatóak általában a legfrissebb tartalmak címei, bevezetői. (Blogok. esetén nem csak bevezető, hanem a teljes tartalom megjelenítése is szokásos ) M Jellemzően felül és bal (esetleg jobb) aldalt megjelenő menűpontok közvetlenül vagy M . Egyre jellemzőbb, hogy többféle navigációs eszközzel is elérhetűnk egyes tartalma. kat, pl. címkék, címkefelhő, kenyérmorzsa menű, stb. 2.3.2. Tartalmak létrehozása . Amennyiben rendelkezünk megfelelő jogosultságokkal, a Navígáció menün megjelenik a Tartalom hozzáadása link (2.21. ábra).
2.3. Tartalmak kezelése 51. oldal Ha üresen hagyjuk, akkor a törzs egy szeletét (kb. 600 karakter) fogja Összegzésnek tekinteni. Szövegformátum A Törzs mező alatt (2.23. ábra) pontos információkat kaphatunk arra nézve, hogy a megadott szöveget hogyan kezelje a Drupal. Az alapértelmezett beállítások a 2.24. ábrán láthatóak, de jelentős eltérés is lehetséges. Ahogy láthatjuk, az alapértelmezett Filtered HTML szövegformátum esetén a linkek kattintható hivatkozások lesznek, és nem kell az a HTML tagot precízen le- írnunk néhány HTML tagot is használhatunk a szövegünk formázására és tagolására, a töb- bi HTML tagot a Drupal eltávolítja a bekezdések tagolását is rábízhatjuk a Drupalra: az üres sor határára precíz bekez- dések jönnek létre a p tag használata nélkül is 2.24. ábra. Összegzés szerkesztése	##### 2.3. Tartalmak kezelése 51. oldal _2.24. ábra. Összegzés szerkesztése_ Ha üresen hagyjuk, akkor a törzs egy szeletét (kb. 600 karakter) fogja Összegzésnek tekinteni. #### Szövegformátum A Törzs mező alatt (2.23. ábra) pontos információkat kaphatunk arra nézve, hogy a megadott szöveget hogyan kezelje a Drupal. Az alapértelmezett beállítások a 2.24. ábrán láthatóak, de jelentős eltérés is lehetséges. Ahogy láthatjuk, az alapértelmezett Filtered HTML szövegformátum esetén  a linkek kattintható hivatkozások lesznek, és nem kell az a HTML tagot precízen le írnunk  néhány HTML tagot is használhatunk a szövegünk formázására és tagolására, a töb bi HTML tagot a Drupal eltávolítja  a bekezdések tagolását is rábízhatjuk a Drupalra: az üres sor határára precíz bekez -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.3. Tartalmak kezelése</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">51. oldal</span></p> <p style="top:513.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha üresen hagyjuk, akkor a törzs egy szeletét (kb. 600 karakter) fogja Összegzésnek tekin-</span></p> <p style="top:526.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">teni.</span></p> <p style="top:547.8pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Szövegformátum</span></b></p> <p style="top:572.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Törzs</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> mező alatt (2.23. ábra) pontos információkat kaphatunk arra nézve, hogy a meg-</span></p> <p style="top:585.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">adott szöveget hogyan kezelje a Drupal. Az alapértelmezett beállítások a 2.24. ábrán látha-</span></p> <p style="top:597.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tóak, de jelentős eltérés is lehetséges.</span></p> <p style="top:616.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ahogy láthatjuk, az alapértelmezett </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Filtered HTML</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szövegformátum esetén</span></p> <p style="top:636.6pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">a linkek kattintható hivatkozások lesznek, és nem kell az </span><span style="font-family:LiberationMono,serif;font-size:9.0pt;color:#000000">a</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> HTML tagot precízen le-</span></p> <p style="top:649.2pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">írnunk</span></p> <p style="top:669.4pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">néhány HTML tagot is használhatunk a szövegünk formázására és tagolására, a töb-</span></p> <p style="top:682.0pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">bi HTML tagot a Drupal eltávolítja</span></p> <p style="top:702.2pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">a bekezdések tagolását is rábízhatjuk a Drupalra: az üres sor határára precíz bekez-</span></p> <p style="top:714.8pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">dések jönnek létre a </span><span style="font-family:LiberationMono,serif;font-size:9.0pt;color:#000000">p</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> tag használata nélkül is</span></p> <p style="top:489.6pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.24. ábra. 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Az alapértelmezett beállítások a 22ó. ábrán látha- tóak, de jelentős eltérés is lehetséges. .Ahogy láthatjuk, az alapértelmezett Filtered HTMI szövegformátum esetén m a Hnkek kattintható hivatkozások lesznek, és nem kell az a HTMU tagot precizen le- irnunk m néhány HTML tagot is használhatunk a szövegünk formázi bi HTML tagot a Drupal eltávolítja m a bekezdések tagolását is rábízhatjuk a Drupalra: az űres sor határára preciz bekez- dések jönnek létre a p tag használata nélkül ís tagolására, a töb-
52. oldal 2. A Drupal felhasználói szemmel Előfordulhat, hogy a 2.24. ábra Szövegformátum választólistája nem jelenik meg, mivel a felhasználónak csak egyféle szövegformátum használatához van jogosultsága. A lehetőségek listája azonban ekkor is látszik. Mindenképpen figyelembe kell azonban venni, hogy a weboldalak szövegformázásának logikája (az eltérő megjelenítési logika miatt) eléggé eltér a hagyományos, papír alapú szövegszerkesztéstől. Ezért egy kicsit el kell felejtenünk a szövegszerkesztőnk papír alapú logikáját, és meg kell tanulnunk, mit is jelent a felbontásfüggetlen tipográﬁa. Előnézet Előfordulhat, hogy a szerkesztés oldal alján a Mentés nem, csak az Előnézet gomb látható. Ez arra utal, hogy az előnézet használata kötelező, csak második lépésben fogjuk megtalálni a Mentés gombot. Előnézet kérése esetén megtekinthetjük (2.25. ábra), milyen lesz az oldalunk, ha véglegesen beküldjük. Ha most elnavigálnánk a szerkesztési oldalról, és nem a Mentés gombra kattintanánk, akkor az eddig bevitt tartalom véglegesen elveszne. Az oldal Bevezető előnézete akkor fog szerephez jutni, ha az éppen beküldés alatt álló tartalom a kezdő oldalon (vagy más hasonló listázó oldalon) is megjelenő tartalom lesz. Általában a Teljes tartalom előnézetével kell elsősorban foglalkoznunk. Az ismét megjelenő szerkesztőben még szükség esetén módosíthatjuk az oldal tartalmát, majd ha kész vagyunk, kattintsunk a Mentés gombra. Ezzel a tartalmunk elkészült, amit a tájékoztató üzenet is megerősít (2.26. ábra). 2.25. ábra. Előnézet megtekintése	##### 52. oldal 2. A Drupal felhasználói szemmel Előfordulhat, hogy a 2.24. ábra Szövegformátum választólistája nem jelenik meg, mivel a felhasználónak csak egyféle szövegformátum használatához van jogosultsága. A lehetőségek listája azonban ekkor is látszik. Mindenképpen figyelembe kell azonban venni, hogy a weboldalak szövegformázásának logikája (az eltérő megjelenítési logika miatt) eléggé eltér a hagyományos, papír alapú szövegszerkesztéstől. Ezért egy kicsit el kell felejtenünk a szövegszerkesztőnk papír alapú logikáját, és meg kell tanulnunk, mit is jelent a felbontásfüggetlen tipográfia. #### Előnézet Előfordulhat, hogy a szerkesztés oldal alján a Mentés nem, csak az Előnézet gomb látható. Ez arra utal, hogy az előnézet használata kötelező, csak második lépésben fogjuk megtalálni a Mentés gombot. Előnézet kérése esetén megtekinthetjük (2.25. ábra), milyen lesz az oldalunk, ha véglegesen beküldjük. Ha most elnavigálnánk a szerkesztési oldalról, és nem a Mentés gombra kattintanánk, akkor az eddig bevitt tartalom véglegesen elveszne. _2.25. ábra. Előnézet megtekintése_ Az oldal Bevezető előnézete akkor fog szerephez jutni, ha az éppen beküldés alatt álló tartalom a kezdő oldalon (vagy más hasonló listázó oldalon) is megjelenő tartalom lesz. Általában a Teljes tartalom előnézetével kell elsősorban foglalkoznunk. Az ismét megjelenő szerkesztőben még szükség esetén módosíthatjuk az oldal tartalmát, majd ha kész vagyunk, kattintsunk a Mentés gombra. Ezzel a tartalmunk elkészült, amit a tájékoztató üzenet is megerősít (2.26. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">52. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:150.2pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">Előfordulhat, hogy a 2.24. ábra </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:9.0pt;color:#000000">Szövegformátum</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> választólistája nem jelenik meg, mivel a felhasználónak csak </span></p> <p style="top:160.5pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">egyféle szövegformátum használatához van jogosultsága. A lehetőségek listája azonban ekkor is látszik.</span></p> <p style="top:177.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Mindenképpen figyelembe kell azonban venni, hogy a weboldalak szövegformázásának lo-</span></p> <p style="top:189.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">gikája (az eltérő megjelenítési logika miatt) eléggé eltér a hagyományos, papír alapú szö-</span></p> <p style="top:202.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">vegszerkesztéstől. Ezért egy kicsit el kell felejtenünk a szövegszerkesztőnk papír alapú logi-</span></p> <p style="top:214.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">káját, és meg kell tanulnunk, mit is jelent a felbontásfüggetlen tipográﬁa.</span></p> <p style="top:236.6pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Előnézet</span></b></p> <p style="top:261.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Előfordulhat, hogy a szerkesztés oldal alján a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Mentés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> nem, csak az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Előnézet</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gomb látható. </span></p> <p style="top:273.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ez arra utal, hogy az előnézet használata kötelező, csak második lépésben fogjuk megtalál-</span></p> <p style="top:286.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ni a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Mentés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombot.</span></p> <p style="top:305.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Előnézet kérése esetén megtekinthetjük (2.25. ábra), milyen lesz az oldalunk, ha véglegesen </span></p> <p style="top:317.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">beküldjük. Ha most elnavigálnánk a szerkesztési oldalról, és nem a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Mentés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombra kattin-</span></p> <p style="top:330.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tanánk, akkor az eddig bevitt tartalom véglegesen elveszne.</span></p> <p style="top:628.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az oldal </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Bevezető előnézete</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> akkor fog szerephez jutni, ha az éppen beküldés alatt álló tar-</span></p> <p style="top:641.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">talom a kezdő oldalon (vagy más hasonló listázó oldalon) is megjelenő tartalom lesz. Álta-</span></p> <p style="top:654.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lában a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Teljes tartalom előnézeté</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">vel kell elsősorban foglalkoznunk.</span></p> <p style="top:672.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az ismét megjelenő szerkesztőben még szükség esetén módosíthatjuk az oldal tartalmát, </span></p> <p style="top:685.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">majd ha kész vagyunk, kattintsunk a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Mentés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombra. Ezzel a tartalmunk elkészült, amit a </span></p> <p style="top:697.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tájékoztató üzenet is megerősít (2.26. ábra).</span></p> <p style="top:605.0pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.25. ábra. 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2. A Drupal felhasználói szemmel Előfondulbat, hogy a 226. áben Szóvegformábn választálátája nem jelenők m; mével a felhasználónak csak egytéle szövegfonmátum haszzálatához van jogosultsága. A lebetőségek tj azoaban ekkor 4 Ktszik Mindenképpen figyelembe kell azonban venni, hogy a weboldalak szövegformázásának lo. gíkója (az eltérő megjelenítési logika miatt) eléggé eltér a hagyományos, papír alapú szó- Vvegszerkesztéstől. Ezért egy kicsit el kell felejtenünk a szövegszerkesztőnk papír alapú logi. káját, és meg kell tanulnunk, mit is jelent a felbontásfüggetlen típográfia. Előnézet Előfordulhat, hogy a szerkesztés oldal alján a Mentés nem, csak az Előnézet gomb látható. Ez arra utal, hogy az előnézet használata kötelező, csak második lépésben fogjuk megtalál. ni a Mentés gombot, Előnézet kérése esetén megtekinthetjük (2.25. ábra), milyen lesz az oldalunk, ha véglegesen beküldjük. Ha most elnavigálnánk a szerkesztési oldalról, és nem a Mentés gombra kattin. tanánk, akkor az eddíg bevítt tartalom véglegesen elveszne. Er Előnézet tt pl Te dővésett bosamról Tdsszmizikezébeen mor én 1nt eeer érvonkt e oo nér e lr ko v a e or eleel kcr té tldtent lo e ml azós j lk pn lk o mg b á tgybken ko á lk pupritrzeztrturrán 225. ábra. Előnézet megtekintése .Az oldal Bevezető előnézete akkor fog szerephez jutni, ha az éppen beküldés alatt álló tar- talom a kezdő oldalon (vagy más hasonló listázó oldalon) is megjelenő tartalom lesz. Ált lában a Teljes tartalom előnézetével kell elsősorban foglalkoznunk. .Az ismét megjelenő szerkesztőben még szükség esetén módosíthatjuk az oldal tartalmát, majd ha kész vagyunk, kattintsunk a Mentés gombra. Ezzel a tartalmunk elkészült, amít a tájékoztató üzenet is megerősít (2.26. ábra).
2.3. Tartalmak kezelése 53. oldal Vizuális szerkesztő Ha az oldal adminisztrátora engedélyezi, akkor lehetőségünk van ún. vizuális szerkesztők (WYSIWYG editor) használatára is. A 2.27. ábrán látszik, hogy a tartalmak bevitele a vizuális szerkesztők segítségével hasonló módon oldható meg, mint ahogy azt a szövegszerkesztőnkben is megszokhattuk. Használatukhoz nem szükséges a HTML alapos ismerete, bár az alapokkal (1.5 fejezet) érdemes tisztában lennünk. A weboldalakon többféle WYSIWYG editor használata is elterjedt. A 2.27. ábrán az FCKEditor44, a szerző kedvence látható. Elterjedt még pl. a TinyMCE45, a YUI editor46 és több másik megoldás. 44 http://ckeditor.com/ 45 http://tinymce.moxiecode.com/ 46 http://developer.yahoo.com/yui/editor/ 2.26. ábra. A beküldött tartalom létrejött 2.27. ábra. WYSIWYG editor	##### 2.3. Tartalmak kezelése 53. oldal _2.26. ábra. A beküldött tartalom létrejött_ #### Vizuális szerkesztő Ha az oldal adminisztrátora engedélyezi, akkor lehetőségünk van ún. vizuális szerkesztők (WYSIWYG editor) használatára is. A 2.27. ábrán látszik, hogy a tartalmak bevitele a vizuális szerkesztők segítségével hasonló módon oldható meg, mint ahogy azt a szövegszerkesztőnkben is megszokhattuk. Használatukhoz nem szükséges a HTML alapos ismerete, bár az alapokkal (1.5 fejezet) érdemes tisztában lennünk. A weboldalakon többféle WYSIWYG editor használata is elterjedt. A 2.27. ábrán az FCKEditor[44], a szerző kedvence látható. Elterjedt még pl. a TinyMCE[45], a YUI editor[46] és több másik megoldás. -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.3. Tartalmak kezelése</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">53. oldal</span></p> <p style="top:361.5pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Vizuális szerkesztő</span></b></p> <p style="top:386.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha az oldal adminisztrátora engedélyezi, akkor lehetőségünk van ún. vizuális szerkesztők </span></p> <p style="top:398.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(WYSIWYG editor) használatára is. A 2.27. ábrán látszik, hogy a tartalmak bevitele a vizu-</span></p> <p style="top:411.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ális szerkesztők segítségével hasonló módon oldható meg, mint ahogy azt a szövegszer-</span></p> <p style="top:423.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kesztőnkben is megszokhattuk. Használatukhoz nem szükséges a HTML alapos ismerete, </span></p> <p style="top:436.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">bár az alapokkal (1.5 fejezet) érdemes tisztában lennünk.</span></p> <p style="top:455.0pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">A weboldalakon többféle WYSIWYG editor használata is elterjedt. A 2.27. ábrán az FCKEditor</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:5.2pt;color:#000000">44</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">, a szerző ked-</span></p> <p style="top:465.2pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">vence látható. Elterjedt még pl. a TinyMCE</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:5.2pt;color:#000000">45</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">, a YUI editor</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:5.2pt;color:#000000">46</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> és több másik megoldás.</span></p> <p style="top:698.6pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">44</span><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c"> http://ckeditor.com/</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:708.8pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">45</span><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c"> http://tinymce.moxiecode.com/</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:719.1pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">46</span><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c"> http://developer.yahoo.com/yui/editor/</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:337.4pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.26. ábra. 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Tartalmak kezelése 53. oldal 9 oé — "Magamról Töbrenallrnedn e le a öl orzezoeó érhötlelő s m Apszz Gyűlke A eee zzérsséozozatán 226. ábra. A beküldött tartalom létrejött Vizuális szerkesztő Ha az oldal adminisztrátora engedélyezi, akkor lehetőségünk van ún. vizuális szerkesztők (WYSIWYG edítor) használatára is. A 2.27. ábrán látszik, hogy a tartalmak bevítele a vízu- ális szerkesztők segítségével hasonló módon oldható meg, mint ahogy azt a szövegszer kesztőnkben is megszokhattuk, Használatukhoz nem szükséges a HTML alapos ismerete, báraz alapokkal (1.5 fejezet) érdemes tisztában lennünk, vence látható Elterjedt még pl. a TinyáMCE, a YUL edntor és tőbb másik megoldáz — forzéve- -W48 o ———] Hogv én kszem vázságs Xral eten gyermeke ehetek Számony éz 5 ezforezzább, ÉT PVSS91— cl ,Soks nem saltam azt a gyolekezetet abol marsdétssánal sgyottadjok öcsérm ter és s2vot to az karöersk oé szolgásn.1938-öan szonösn megyalásam a ckér Szzoszs Cyulokzzetet 290 szolgál lsaltdom tektér fönlőséget vátotam) 6 ta96: leségemt ár ét ook Ssr és Á, 6t ét lk Oa 65 J sssskemén edokota cb karános mformatka vésstében tanftok elsósorban Jr C/C . erograozst, öttejeszstst Tarvomkezolén Ue Ezen tn a ar honlaojátszerkesztem akzban, 2008 éra egyér váalkozás formában v2zslot Orapal a melfeesztet ét zze kapczoltos öltskst 227. ábra. 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54. oldal 2. A Drupal felhasználói szemmel Érdemes azonban figyelembe venni, hogy egy weboldal – eltérően egy nyomtatásra szánt, szövegszerkesztőben készített dokumentumtól, – akár minden látogató esetén máshogy fog kinézni. Ezért érdemes csupán alapvető formázási tevékenységre szorítkozni. (Egy jól beállított weboldal esetén csak az engedélyezett elemeknek megfelelő gombok használhatók a vizuális szerkesztőn.) A 2.4 fejezetben vissza fogunk térni a vizuális szerkesztők használatára. További információk megadása Bizonyos esetben a címen és a törzsön kívül további információk megadására is van lehetőség. Néhány eset ezek közül: Ha van jogunk hozzá, a tartalmat valamelyik menübe is elhelyezhetjük (2.28. ábra). Fórumtéma beküldése esetén (2.29. ábra) kiválaszthatjuk, hogy melyik fórumhoz tartozzon. 2.28. ábra. Tartalom menübe helyezése	##### 54. oldal 2. A Drupal felhasználói szemmel Érdemes azonban figyelembe venni, hogy egy weboldal – eltérően egy nyomtatásra szánt, szövegszerkesztőben készített dokumentumtól, – akár minden látogató esetén máshogy fog kinézni. Ezért érdemes csupán alapvető formázási tevékenységre szorítkozni. (Egy jól beállított weboldal esetén csak az engedélyezett elemeknek megfelelő gombok használhatók a vizuális szerkesztőn.) A 2.4 fejezetben vissza fogunk térni a vizuális szerkesztők használatára. #### További információk megadása Bizonyos esetben a címen és a törzsön kívül további információk megadására is van lehetőség. Néhány eset ezek közül: Ha van jogunk hozzá, a tartalmat valamelyik menübe is elhelyezhetjük (2.28. ábra). _2.28. ábra. Tartalom menübe helyezése_ _Fórumtéma beküldése esetén (2.29. ábra) kiválaszthatjuk, hogy melyik fórumhoz tartozzon._ -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">54. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:147.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Érdemes azonban figyelembe venni, hogy egy weboldal – eltérően egy nyomtatásra szánt, </span></p> <p style="top:159.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szövegszerkesztőben készített dokumentumtól, – akár minden látogató esetén máshogy fog </span></p> <p style="top:172.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kinézni. Ezért érdemes csupán alapvető formázási tevékenységre szorítkozni. (Egy jól beál-</span></p> <p style="top:185.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lított weboldal esetén csak az engedélyezett elemeknek megfelelő gombok használhatók a </span></p> <p style="top:197.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">vizuális szerkesztőn.)</span></p> <p style="top:216.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A 2.4 fejezetben vissza fogunk térni a vizuális szerkesztők használatára.</span></p> <p style="top:238.1pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">További információk megadása</span></b></p> <p style="top:262.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Bizonyos esetben a címen és a törzsön kívül további információk megadására is van lehe-</span></p> <p style="top:275.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tőség. Néhány eset ezek közül:</span></p> <p style="top:294.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha van jogunk hozzá, a tartalmat valamelyik </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">menübe</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> is elhelyezhetjük (2.28. ábra).</span></p> <p style="top:578.7pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Fórumtéma</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> beküldése esetén (2.29. ábra) kiválaszthatjuk, hogy melyik fórumhoz tartozzon.</span></p> <p style="top:554.8pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.28. ábra. 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A Drupal felhasználói szemmel Érdemes azonban figyelembe venni, hogy egy webaldal - eltérően egy nyomtatásra szánt, szövegszerkesztőben készített dokumentumtól, - akár minden látogató esetén máshogy fog. kínézni. Ezért érdemes csupán alapvető formázási tevékenységre szorítkozni. (Egy jól beál litott sveboldal esetén csak az engedélyezett elemeknek megfelelő gombok használhatók a vázuális szerkesztőn) A 2. fejezetben vissza fogunk térni a vizuális szerkesztők haszni "További információk megadása Bizonyos esetben a címen és a törzsön kívül további infori tőség. Néhány eset ezek közül: iók megadására is van lehe. Ha van jogunk hoz lmat valamelyik mendűbe is elhelyezhetjük (228. ábra). Menüübeállítások. — Z Menpostothozléte Menüpontness agznról okeás aldoc jlenlk s ha a egór a mendborztszás £ ortl, Szülődem -efőmenés [2] n. AA osebb szört menőgontok a nebezeté hrztkozások előt jleszek m 228. ábra. Tartalom menübe helyezése Fórumtéma beküldése esetén (2.29. ábra) kiválaszthatjuk, hogy melyik főrumhoz tartozzon.
56. oldal 2. A Drupal felhasználói szemmel Ezek természetesen nem csak információt hordoznak, hanem navigációs lehetőséget is adnak: a címke feliratára kattintva az ugyanezen címkével ellátott tartalmak listázhatóak. Egyes esetekben (tartalomtípustól és jogosultságoktól függően) a tartalom mellékleteként csatolt állományok is alkalmazhatók. A melléklet állományokra nézve méret- és típuskorlátozás lehet érvényben. Az állomány helyét és nevét a Tallózás gombbal adhatjuk meg (2.32. ábra). A Feltöltés gomb elvégzi a tényleges feltöltést. A weben érdemes ékezetes karakterek és írásjelen nélkül elnevezett állományokkal dolgozni. Egyes esetekben (pl. a weboldal költöztetése egy másik tárhelyszolgáltatóhoz) problémás lehet a speciális karakterekkel. A 13.15fejezetben látni fogjuk, hogy a Drupal képes helyettünk figyelni erre a szempontra. A Feltöltés után a fájlnévnél beszédesebb Leírást is megadhatunk (2.33. ábra). Ha szükséges, kikapcsolhatjuk az állomány Megjelenítését. Így a fájl ugyan a webszerverre kerül, de nem lesz hozzá automatikusan letöltési link gyártva. Végül érdemes megemlíteni, hogy akár újabb fájlokat is feltölthetünk: a feltöltés után újabb felület jelenik meg a feltöltött állomány adatai alatt (2.33. ábra). 2.31. ábra. Címkék megjelenése 2.32. ábra. Csatolmány elhelyezése	##### 56. oldal 2. A Drupal felhasználói szemmel _2.31. ábra. Címkék megjelenése_ Ezek természetesen nem csak információt hordoznak, hanem navigációs lehetőséget is adnak: a címke feliratára kattintva az ugyanezen címkével ellátott tartalmak listázhatóak. Egyes esetekben (tartalomtípustól és jogosultságoktól függően) a tartalom mellékleteként csatolt állományok is alkalmazhatók. A melléklet állományokra nézve méret- és típuskorlátozás lehet érvényben. Az állomány helyét és nevét a Tallózás gombbal adhatjuk meg (2.32. ábra). A Feltöltés gomb elvégzi a tényleges feltöltést. A weben érdemes ékezetes karakterek és írásjelen nélkül elnevezett állományokkal dolgozni. Egyes esetekben (pl. a weboldal költöztetése egy másik tárhelyszolgáltatóhoz) problémás lehet a speciális karakterekkel. A 13.15fejezetben látni fogjuk, hogy a Drupal képes helyettünk figyelni erre a szempontra. _2.32. ábra. Csatolmány elhelyezése_ A Feltöltés után a fájlnévnél beszédesebb Leírást is megadhatunk (2.33. ábra). Ha szükséges, kikapcsolhatjuk az állomány Megjelenítését. Így a fájl ugyan a webszerverre kerül, de nem lesz hozzá automatikusan letöltési link gyártva. Végül érdemes megemlíteni, hogy akár újabb fájlokat is feltölthetünk: a feltöltés után újabb felület jelenik meg a feltöltött állomány adatai alatt (2.33. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">56. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:321.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ezek természetesen nem csak információt hordoznak, hanem navigációs lehetőséget is ad-</span></p> <p style="top:333.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nak: a címke feliratára kattintva az ugyanezen címkével ellátott tartalmak listázhatóak.</span></p> <p style="top:352.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Egyes esetekben (tartalomtípustól és jogosultságoktól függően) a tartalom mellékleteként </span></p> <p style="top:364.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">csatolt állományok is alkalmazhatók. A melléklet állományokra nézve méret- és típuskor-</span></p> <p style="top:377.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">látozás lehet érvényben.</span></p> <p style="top:396.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az állomány helyét és nevét a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Tallózás</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombbal adhatjuk meg (2.32. ábra). A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Feltöltés </span></i></p> <p style="top:408.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">gomb elvégzi a tényleges feltöltést.</span></p> <p style="top:427.2pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">A weben érdemes ékezetes karakterek és írásjelen nélkül elnevezett állományokkal dolgozni. Egyes esetekben </span></p> <p style="top:437.5pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">(pl. a weboldal költöztetése egy másik tárhelyszolgáltatóhoz) problémás lehet a speciális karakterekkel. A 13.15-</span></p> <p style="top:447.7pt;left:85.2pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">fejezetben látni fogjuk, hogy a Drupal képes helyettünk figyelni erre a szempontra.</span></p> <p style="top:591.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Feltöltés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> után a fájlnévnél beszédesebb </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Leírást</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> is megadhatunk (2.33. ábra).</span></p> <p style="top:610.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha szükséges, kikapcsolhatjuk az állomány </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Megjelenítés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ét. Így a fájl ugyan a webszerverre </span></p> <p style="top:622.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kerül, de nem lesz hozzá automatikusan letöltési link gyártva.</span></p> <p style="top:641.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Végül érdemes megemlíteni, hogy akár újabb fájlokat is feltölthetünk: a feltöltés után </span></p> <p style="top:654.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">újabb felület jelenik meg a feltöltött állomány adatai alatt (2.33. ábra).</span></p> <p style="top:297.1pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.31. ábra. 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A Drupal felhasználói szemmel Elindult a honlap fejlesztése tp8 B horiloska fée kérködőedben. 231. ábra. Címkék megjelenése Ezek természetesen nem csak információt hordoznak, hanem navigációs lehetőséget is ad. nak: a cimke feliratára kattintva az ugyanezen címkével ellátott tartalmak listázhatóak. Egyes esetekben (tartalomtípustól és jogosultságoktól függően) a tartalom mellékleteként csatolt állományok is alkalmazhatók. A melléklet állományokra nézve méret- és típuskor- látozás lehet érvényben. .Az állomány helyét és nevét a Tallózás gombbal adhatjuk meg (232. ábra). A Feltőltés "Bomb elvégzi a tényleges feltöltést. A wben érdenes élezetes kazakterek é írásjelen nélkül elsevezett állormányokkal dolgozni. Egyes esetelben Á a mebeldal költéztetése egy másik tárhelyszolgáltatóbsz) problémás lehet a speciáls kazaloerekkel A 13. ijezetben látni fogjak. hogy a Dragpal képes helyeztünk Hgyelni err a szenporaia. — jfájl hozzásdása E A jok ét ngejető a MI fznóálnt Bzipssokdoc:ds ot ods olt odp páí l9. 232 ábra. Csatolmány elhelyezése A Feltöltés után a fájlnévnél beszédesebb Lzírástis megadhatunk (2.33. ábra). Ha szükséges, kikapcsolhatjuk az állomány Megjelenítését. Így a fájl ugyan a webszerverre kerül, de nem lesz hozzá automatikusan letöltési link gyártva. Végül érdemes megesnlíteni, hogy akár újabb fíjlokat is feltöltbetűnk: a feltöltés után újabb felület jelenik meg a feltőltőtt állomány adatai alatt (2.33. ábra).
2.3. Tartalmak kezelése 57. oldal A beküldés után a csatolt állományok letölthetővé válnak (2.34. ábra). Egyelőre nem foglalkozunk azzal a kérdéssel, hogy az adott oldal hol (pl. milyen menüpontban) lesz elérhető a honlapunkon. 2.3.3. Tartalom szerkesztése, törlése Ha később visszalátogatunk az előzőleg létrehozott oldalunkra, akkor az oldal címe mellett az aktuális Megtekintés fül mellett a Szerkesztés fület is megﬁgyelhetjük (2.31. ábra). A Szerkesztés fülön a beküldéshez hasonlóan módosítani vagy akár törölni tudjuk a tartalmunkat. Persze előfordulhat, hogy szerkeszteni van, de törölni nincs jogunk. Ilyenkor a megfelelő gomb se lesz látható a felhasználó számára. 2.33. ábra. Csatolmány finomítása, újabb csatolmányok felvitele 2.34. ábra. Letölthető csatolmány	##### 2.3. Tartalmak kezelése 57. oldal _2.33. ábra. Csatolmány finomítása, újabb csatolmányok felvitele_ A beküldés után a csatolt állományok letölthetővé válnak (2.34. ábra). _2.34. ábra. Letölthető csatolmány_ Egyelőre nem foglalkozunk azzal a kérdéssel, hogy az adott oldal hol (pl. milyen menüpontban) lesz elérhető a honlapunkon. ### 2.3.3. Tartalom szerkesztése, törlése Ha később visszalátogatunk az előzőleg létrehozott oldalunkra, akkor az oldal címe mellett az aktuális Megtekintés fül mellett a Szerkesztés fület is megfigyelhetjük (2.31. ábra). A Szerkesztés fülön a beküldéshez hasonlóan módosítani vagy akár törölni tudjuk a tartalmunkat. Persze előfordulhat, hogy szerkeszteni van, de törölni nincs jogunk. Ilyenkor a megfelelő gomb se lesz látható a felhasználó számára. -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.3. Tartalmak kezelése</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">57. oldal</span></p> <p style="top:375.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A beküldés után a csatolt állományok letölthetővé válnak (2.34. ábra).</span></p> <p style="top:546.0pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">Egyelőre nem foglalkozunk azzal a kérdéssel, hogy az adott oldal hol (pl. milyen menüpontban) lesz elérhető a </span></p> <p style="top:556.2pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">honlapunkon.</span></p> <p style="top:582.1pt;left:113.6pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.3.3. Tartalom szerkesztése, törlése</span></b></p> <p style="top:614.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha később visszalátogatunk az előzőleg létrehozott oldalunkra, akkor az oldal címe mellett </span></p> <p style="top:627.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">az aktuális </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Megtekintés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> fül mellett a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Szerkesztés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> fület is megﬁgyelhetjük (2.31. ábra).</span></p> <p style="top:645.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Szerkesztés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> fülön a beküldéshez hasonlóan módosítani vagy akár törölni tudjuk a tartal-</span></p> <p style="top:658.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">munkat. </span></p> <p style="top:677.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Persze előfordulhat, hogy szerkeszteni van, de törölni nincs jogunk. Ilyenkor a megfelelő </span></p> <p style="top:689.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">gomb se lesz látható a felhasználó számára.</span></p> <p style="top:351.7pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.33. ábra. 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Fenönés A fok ár lzídlébb s aD ebet parnáetzt elzgezzk docais po ods ot ody pá á 233. ábra. Csatolmány finomitása, újabb csatolmányok felvitele A beküldés után a csatolt állományok letölthetővé válnak (2.34. ábra). A Kocskeznét Pőiskola GAM Kazának laformatika Intézetében tanítok elsőzorban Java, /C e programozást, Webfejlesztést, Tartalomkezelést, Lincsot. Ezen kívül a kaz honlapját. szechesztem aktívan. 2008 óta egyén vállalkozási formában vállalok Drupal alapó vobfejlesztést, és szzel kapcsolatos óktatást. Csatelmány: ElDrapal 7 alapiszmoretek 236. ábra. Letölthető csatolmány Egyelőre nem fogdalkozank azzal a kérdésel,hogy az adott oldal hol (pl milyen menüpoztban) lesz elérbető a 2.3.3. Tartalom szerkesztése, törlése Ha később visszalátogatunk az előzöleg létrehozott oldalunkra, akkor az oldal címe mellett az aktuális Megfekintés fül mellett a Szerkesztés filet is megfigyelhetjük (231. ábra). A Szerkesztés fülön a beküldéshez hasonlóan módosítani vagy akár törölni tudjuk a tartal- munkat. Persze előfordulhat, hogy szerkeszteni van, de törölni nincs jogunk. Ilyenkor a megfelelő. gomb se lesz látható a felhasználó számára.
58. oldal 2. A Drupal felhasználói szemmel A tartalom törlése nem visszavonható művelet! Ezért inkább a tartalom elrejtését szokás végezni a tényleges törlés helyett. Változatok kezelése A Drupal lehetőséget ad arra, hogy egy tartalom szerkesztésekor és újbóli mentésekor ne írjuk felül az előző változatot, hanem – mintegy biztonsági mentést – megőrizzük. Így előfordulhat, hogy egy tartalomnak több tucat változatát is őrzi a weboldalunk. Ha az adminisztrátor beállította a változatok kezelését, akkor szerkesztéskor újabb eszközöket láthatunk (2.35. ábra). Ha van jogunk a változatok megtekintésére, akkor a Mentés után egy harmadik Változatok fül is megjelenik a tartalom címe alatt (2.36. ábra). 2.35. ábra. Változatok kezelése	##### 58. oldal 2. A Drupal felhasználói szemmel _A tartalom törlése nem visszavonható művelet! Ezért inkább a tartalom elrejtését szokás_ végezni a tényleges törlés helyett. #### Változatok kezelése A Drupal lehetőséget ad arra, hogy egy tartalom szerkesztésekor és újbóli mentésekor ne írjuk felül az előző változatot, hanem – mintegy biztonsági mentést – megőrizzük. Így előfordulhat, hogy egy tartalomnak több tucat változatát is őrzi a weboldalunk. Ha az adminisztrátor beállította a változatok kezelését, akkor szerkesztéskor újabb eszközöket láthatunk (2.35. ábra). _2.35. ábra. Változatok kezelése_ Ha van jogunk a változatok megtekintésére, akkor a Mentés után egy harmadik Változatok fül is megjelenik a tartalom címe alatt (2.36. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">58. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:150.4pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">A tartalom törlése nem visszavonható művelet!</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> Ezért inkább a tartalom elrejtését szokás </span></p> <p style="top:163.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">végezni a tényleges törlés helyett.</span></p> <p style="top:184.8pt;left:85.2pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Változatok kezelése</span></b></p> <p style="top:209.5pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A Drupal lehetőséget ad arra, hogy egy tartalom szerkesztésekor és újbóli mentésekor ne </span></p> <p style="top:222.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">írjuk felül az előző változatot, hanem – mintegy biztonsági mentést – megőrizzük. Így elő-</span></p> <p style="top:234.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">fordulhat, hogy egy tartalomnak több tucat változatát is őrzi a weboldalunk.</span></p> <p style="top:253.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha az adminisztrátor beállította a változatok kezelését, akkor szerkesztéskor újabb eszkö-</span></p> <p style="top:265.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">zöket láthatunk (2.35. ábra).</span></p> <p style="top:432.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha van jogunk a változatok megtekintésére, akkor a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Mentés</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> után egy harmadik </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Változatok </span></i></p> <p style="top:445.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">fül is megjelenik a tartalom címe alatt (2.36. ábra).</span></p> <p style="top:408.8pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.35. ábra. 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A Drupal felhasználói szemmel A tartalom törlése nem visszavonható művelet! Ezért inkább a tartalom elrejtését szokás. "végezni a tényleges törlés helyett. Változatok kezelése ,A Drupal lehetőséget ad arra, hogy egy tartalom szerkesztésekor és újbóli mentésekor ne. írjuk felül az előző változatot, hanem - mintegy biztonsági mentést - megőrizzük. Így elő. fordulhat, hogy egy tartalomnak tőbb tucat változatát is őzi a weboldalunk. Ha az adminisztrátor beállította a változatok kezelését, akkor szerkesztéskor újabb eszkö- zöket láthatunk (235. ábra). —Menübeállítások — Változataapló üzenet va zt a megjegyzést csak a szerkesztők táják maj változatinformáció Oíváltozzt p magaráztot bítostaz evégze mzódosíásolsz A v szezőt lc né 235. ábra. Változatok kezelése Ha van jogunk a változatok megtekintésére, akkor a Mentés után egy harmadik Változafok fül is megjelenik a tartalom címe alatt (236. ábra).
2.3. Tartalmak kezelése 59. oldal A Változat oszlopban megtekinthetjük, és – ha jogunk van – visszaállíthatunk egy korábbi állapotot a visszaállítás link segítségével. Ekkor a korábbi változatról egy újabb másolat készül, amit egyből szerkeszthetünk is. 2.4. A vizuális szerkesztők használata Ahogy a 2.3.2 fejezetben már láttuk, a WYSIWYG editorok kényelmesebbé teszik a tartalmaink szerkesztését. Ebben a fejezetben néhány egyszerű példán keresztül fogjuk megnézni az alapfunkciókat. Mindezt olyan megközelítéssel tesszük, amely – a szerző véleménye szerint – a legalkalmasabb kezdők számára. Drupal esetén tucatnyi editor közül választhatunk. Most a legelterjedtebb CKEditort47 fogjuk használni. 2.4.1. Alapelv A webes vizuális szerkesztők lényegében annyit tesznek, hogy a HTML forráskód szerkesztést elfedik előlünk. Ezzel a szerkesztés kényelmesebbé, gyorsabbá válik. De tudnunk kell azt is, hogy a vizuális szerkesztők lehetőségei korlátozottak. Előfordulhatnak olyan szituációk, amikor nem a legideálisabban, sőt rosszul működnek. Ilyenkor kézzel, a szerkesztő kiiktatásával oldhatjuk meg a problémákat. 47 http://ckeditor.com/ 2.36. ábra. Változatok megtekintése	##### 2.3. Tartalmak kezelése 59. oldal _2.36. ábra. Változatok megtekintése_ A Változat oszlopban megtekinthetjük, és – ha jogunk van – visszaállíthatunk egy korábbi állapotot a visszaállítás link segítségével. Ekkor a korábbi változatról egy újabb másolat készül, amit egyből szerkeszthetünk is. ## 2.4. A vizuális szerkesztők használata Ahogy a 2.3.2 fejezetben már láttuk, a WYSIWYG editorok kényelmesebbé teszik a tartalmaink szerkesztését. Ebben a fejezetben néhány egyszerű példán keresztül fogjuk megnézni az alapfunkciókat. Mindezt olyan megközelítéssel tesszük, amely – a szerző véleménye szerint – a legalkalmasabb kezdők számára. Drupal esetén tucatnyi editor közül választhatunk. Most a legelterjedtebb CKEditort[47] fogjuk használni. ### 2.4.1. Alapelv A webes vizuális szerkesztők lényegében annyit tesznek, hogy a HTML forráskód szerkesztést elfedik előlünk. Ezzel a szerkesztés kényelmesebbé, gyorsabbá válik. De tudnunk kell azt is, hogy a vizuális szerkesztők lehetőségei korlátozottak. Előfordulhatnak olyan szituációk, amikor nem a legideálisabban, sőt rosszul működnek. Ilyenkor kézzel, a szerkesztő kiiktatásával oldhatjuk meg a problémákat. -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.3. Tartalmak kezelése</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">59. oldal</span></p> <p style="top:389.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Változat</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> oszlopban megtekinthetjük, és – ha jogunk van – visszaállíthatunk egy korábbi </span></p> <p style="top:402.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">állapotot a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">visszaállítás</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> link segítségével. Ekkor a korábbi változatról egy újabb másolat </span></p> <p style="top:414.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">készül, amit egyből szerkeszthetünk is.</span></p> <p style="top:449.2pt;left:113.6pt;line-height:20.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:20.0pt;color:#000000">2.4. A vizuális szerkesztők használata</span></b></p> <p style="top:490.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ahogy a 2.3.2 fejezetben már láttuk, a WYSIWYG editorok kényelmesebbé teszik a tartal-</span></p> <p style="top:502.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">maink szerkesztését. Ebben a fejezetben néhány egyszerű példán keresztül fogjuk megnéz-</span></p> <p style="top:515.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ni az alapfunkciókat. Mindezt olyan megközelítéssel tesszük, amely – a szerző véleménye </span></p> <p style="top:527.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szerint – a legalkalmasabb kezdők számára.</span></p> <p style="top:546.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Drupal esetén tucatnyi editor közül választhatunk. Most a legelterjedtebb CKEditort</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">47</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> fog-</span></p> <p style="top:559.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">juk használni.</span></p> <p style="top:587.1pt;left:113.6pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.4.1. Alapelv</span></b></p> <p style="top:619.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A webes vizuális szerkesztők lényegében annyit tesznek, hogy a HTML forráskód szer-</span></p> <p style="top:632.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kesztést elfedik előlünk. Ezzel a szerkesztés kényelmesebbé, gyorsabbá válik.</span></p> <p style="top:650.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">De tudnunk kell azt is, hogy a vizuális szerkesztők lehetőségei korlátozottak. Előfordulhat-</span></p> <p style="top:663.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nak olyan szituációk, amikor nem a legideálisabban, sőt rosszul működnek. Ilyenkor kéz-</span></p> <p style="top:676.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">zel, a szerkesztő kiiktatásával oldhatjuk meg a problémákat.</span></p> <p style="top:719.1pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">47</span><span style="font-family:Calibri,serif;font-size:8.0pt;color:#4c4c4c"> http://ckeditor.com/</span><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000"> </span></p> <p style="top:365.9pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.36. ábra. 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AgBgaNBWg0daJTBGWwAADA3aavBIqwTGaAsAgKFBWw0eaZXAOLm2AAAAgPni2LFj aQNEWwAAAEBK0BYAAACQErQFAAAApARtAQAAAClBWwAAAEBK0BYAAACQErQFAAAA pARtAQAAAClBWwAAAEBK/j/jgv5aCNhB5wAAAABJRU5ErkJggg=="> </div>	page_354.png	59. oldal Elindult a honlap fejlesztése változatai pizgyetámés Szatoszós : Vátozzok A változatóklehetévé teszlka tartaksak bülöcböző változatai közötő eltérések követésétésa. assztéeéstegy korábáó változathoz, pr ] é 236. ábra. Változatok megtekintése A Változat oszlopban megtekinthetjük, és - ha jogunk van - visszaállíthatunk egy korábbi állapotot a visszaállítás ink segítségével. Ekkor a korábbi változatról egy újabb másolat készül, amit egyből szerkeszthetünk is. 2.4. A vizuális szerkesztők használata Ahogy a 23.2 fejezetben már láttuk, a WYSIWYG edítorok kényelmesebbé teszik a tartal- maink szerkesztését. Ebben a fejezetben néhány egyszerű példán keresztül fogjuk megnéz- ni az alapfunkciókat. Mindezt olyan megközelítéssel tesszük, amely - a szerző véleménye szerint - a legalkalmasabb kezdők számára. Drupal esetén tucatnyi edítor közül választhatunk. Most a legelterjedtebb CKEdítort:" fog- juk használni 2.4.1. Alapelv A swebes vizuális szerkesztők lényegében annyit tesznek, hogy a HTMI. forráskód szer- kesztést elfedik előlűnk. Ezzel a szerkesztés kényelmesebbé, gyorsabbá válik. De tudnunk kell azt is, hogy a vizuális szerkesztők lehetőségei korlátozottak. Előfordulhat- mak olyan szítuációk, amikor nem a legidcálisabban, sőt rosszul működnek. flyenkor kéz- zel, a szerkesztő kiiktatásával oldhatjuk meg a problémákat. o]
60. oldal 2. A Drupal felhasználói szemmel Az 1.2.1 fejezetben már láttuk a webes tipográfia korlátait is. Ha ehhez még hozzávesszük, hogy egy weboldal esetén rendkívül fontos az egyes oldalak egységes megjelenése is, akkor a vizuális szerkesztőt használó tartalomszerkesztők számára elég korlátozott lehetőségeket szabad és kell nyújtani. Képzeljük el, milyen benyomást keltene az a weboldal, ahol az egyik tartalomszerkesztő minden szöveget középre igazít, óriási betűket alkalmaz minden második mondatban, és tucatnyi színt használ a fontosabb szavak kiemelésére, míg a másik tartalomszerkesztő a nagybetűs írásmódot, és a sorkizárt írásmódot favorizálja. Ha ilyen szabadságot adunk a szerkesztőinknek, akkor nem lehet egységes és igényes látványt kialakítani az oldalunkon. A szerző véleménye szerint egy hír vagy blogbejegyzés szerkesztéséhez kb. a következő funkciókat szabad megengedni a laikus tartalomszerkesztőknek: félkövér és dőlt formázás a soron belüli kiemelésre felsorolás és számozás, akár több szinten egymásba ágyazva kifelé mutató, vagy a honlapon belül maradó linkek kép a folyó szövegben való illusztrációként a folyó bekezdések és sortörések mellett különböző szintű címek	##### 60. oldal 2. A Drupal felhasználói szemmel Az 1.2.1 fejezetben már láttuk a webes tipográfia korlátait is. Ha ehhez még hozzávesszük, hogy egy weboldal esetén rendkívül fontos az egyes oldalak egységes megjelenése is, akkor a vizuális szerkesztőt használó tartalomszerkesztők számára elég korlátozott lehetőségeket szabad és kell nyújtani. Képzeljük el, milyen benyomást keltene az a weboldal, ahol az egyik tartalomszerkesztő minden szöveget középre igazít, óriási betűket alkalmaz minden második mondatban, és tucatnyi színt használ a fontosabb szavak kiemelésére, míg a másik tartalomszerkesztő a nagybetűs írásmódot, és a sorkizárt írásmódot favorizálja. Ha ilyen szabadságot adunk a szerkesztőinknek, akkor nem lehet egységes és igényes látványt kialakítani az oldalunkon. A szerző véleménye szerint egy hír vagy blogbejegyzés szerkesztéséhez kb. a következő funkciókat szabad megengedni a laikus tartalomszerkesztőknek:  félkövér és dőlt formázás a soron belüli kiemelésre  felsorolás és számozás, akár több szinten egymásba ágyazva  kifelé mutató, vagy a honlapon belül maradó linkek  kép a folyó szövegben való illusztrációként  a folyó bekezdések és sortörések mellett különböző szintű címek -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">60. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:147.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Az 1.2.1 fejezetben már láttuk a webes tipográfia korlátait is. Ha ehhez még hozzávesszük, </span></p> <p style="top:159.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">hogy egy weboldal esetén rendkívül fontos az egyes oldalak egységes megjelenése is, ak-</span></p> <p style="top:172.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kor a vizuális szerkesztőt használó tartalomszerkesztők számára elég korlátozott lehetősé-</span></p> <p style="top:185.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">geket szabad és kell nyújtani.</span></p> <p style="top:203.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Képzeljük el, milyen benyomást keltene az a weboldal, ahol az egyik tartalomszerkesztő </span></p> <p style="top:216.3pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">minden szöveget középre igazít, óriási betűket alkalmaz minden második mondatban, és </span></p> <p style="top:228.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">tucatnyi színt használ a fontosabb szavak kiemelésére, míg a másik tartalomszerkesztő a </span></p> <p style="top:241.4pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nagybetűs írásmódot, és a sorkizárt írásmódot favorizálja. Ha ilyen szabadságot adunk a </span></p> <p style="top:253.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szerkesztőinknek, akkor nem lehet egységes és igényes látványt kialakítani az oldalunkon.</span></p> <p style="top:272.7pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A szerző véleménye szerint egy hír vagy blogbejegyzés szerkesztéséhez kb. a következő </span></p> <p style="top:285.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">funkciókat szabad megengedni a laikus tartalomszerkesztőknek:</span></p> <p style="top:305.5pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">félkövér és dőlt formázás a soron belüli kiemelésre</span></p> <p style="top:325.7pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">felsorolás és számozás, akár több szinten egymásba ágyazva</span></p> <p style="top:346.0pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kifelé mutató, vagy a honlapon belül maradó linkek</span></p> <p style="top:366.2pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kép a folyó szövegben való illusztrációként</span></p> <p style="top:386.5pt;left:99.3pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">a folyó bekezdések és sortörések mellett különböző szintű címek</span></p> </div>	page_355.png	60. oldal 2. A Drupal felhasználói szemmel Az 1.2. fejezetben már láttuk a webes tipográfia kozlátait is. Ha ehhez még hozzávesszük, hogy egy weboldal esetén rendkívül fontos az egyes oldalak egységes megjelenése is, ak- kor a vizuilis szerkesztőt használó tartalomszerkesztők számára elég korlátozott lehetősé. aeket szabad és kell nyújtani. Képzeljük el, milyen benyomást keltene az a wcboldal, ahol az egyík tartalomszerkesztő minden szöveget középre igazít, óriási betűket alkalmaz minden második mondatban, és tucatnyi színt használ a fontosabb szavak kiemelésére, míg a másik tartalomszerkesztő a magybetűs írásmódot, és a sorkizárt írásmódot favorizálja. Ha ilyen szabadságot adunk a szerkesztőinknek, akkor nem lehet egységes és igényes látványt kialakítani az oldalunkon. A szerző véleménye szerint egy hír vagy blogbejegyzés szerkesztéséhez kb. a következő funkciókat szabad megengedni a laikus tartalomszerkesztöknek; m félkövér és dölt formázás a soron belüli kiemelésre m . felsorolás és számozás, akár több szinten egymácba ágyazva a . kifelé mutató, vagy a honlapon belül maradó linkek m . képa fölyó szövegben való illusztrációként a folyó bekezdések és sortörések mellett különböző szintű címek
2.4. A vizuális szerkesztők használata 61. oldal Érdemes megemlíteni néhány funkciót, amit a szerző szándékosan nem szokott engedélyezni tartalomszerkesztők számára. A weboldal egységes látványvilága miatt nem javasolt: balra, középre és jobbra igazítás térközök, behúzások színes betűk és hátterek betűtípusok kisebb és nagyobb betűk bekezdések közötti elválasztó vonalak képek balra, jobbra igazítása vegyesen Tipográfiai okok miatt nem javasolt: sorkizárt igazítás (az elválasztás hiánya miatt) aláhúzás (a linkekkel való összekeverhetőség, és az olvashatóság jelentős romlása miatt) a csupa nagybetűs, szóközzel ritkított írásmód (ezek az írógépes korszak maradványai, erre jobb megoldás a dőlt, és félkövér formázás konzekvens alkalmazása) táblázatok (ami papíron kifér és olvasható, az egy weboldalon általában használhatatlan; kisebb táblázatokat pedig sokszor inkább felsorolásként érdemes48 közzétenni) képek körbefuttatása, speciális esetektől eltekintve (általában sokkal kevesebb hely van a nyomtatott médiához képest) 2.4.2. Ajánlott módszer A vizuális szerkesztőket – a szerző véleménye szerint – a következő módszerrel érdemes használni. 1. írjuk meg helyben, vagy illesszük be a nyers szöveget 2. állítsuk be a szöveg struktúráját (kevésbé szerencsés megfogalmazás a formázás) Először is nézzük meg, mit szeretnénk elérni, mi lesz a végeredmény (2.37. ábra). 48 A szerző e mondatok írása közben is először táblázatot akart alkalmazni, de inkább a felsorolások, sortörések és zárójelek alkalmazása mellett döntött.	##### 2.4. A vizuális szerkesztők használata 61. oldal Érdemes megemlíteni néhány funkciót, amit a szerző szándékosan nem szokott engedélyezni tartalomszerkesztők számára. #### A weboldal egységes látványvilága miatt nem javasolt:  balra, középre és jobbra igazítás  térközök, behúzások  színes betűk és hátterek  betűtípusok  kisebb és nagyobb betűk  bekezdések közötti elválasztó vonalak  képek balra, jobbra igazítása vegyesen #### Tipográfiai okok miatt nem javasolt:  sorkizárt igazítás (az elválasztás hiánya miatt)  aláhúzás (a linkekkel való összekeverhetőség, és az olvashatóság jelentős romlása miatt)  a csupa nagybetűs, szóközzel ritkított írásmód (ezek az írógépes korszak maradványai, erre jobb megoldás a dőlt, és félkövér formázás konzekvens alkalmazása)  táblázatok (ami papíron kifér és olvasható, az egy weboldalon általában használhatatlan; kisebb táblázatokat pedig sokszor inkább felsorolásként érdemes[48] közzétenni)  képek körbefuttatása, speciális esetektől eltekintve (általában sokkal kevesebb hely van a nyomtatott médiához képest) ### 2.4.2. Ajánlott módszer A vizuális szerkesztőket – a szerző véleménye szerint – a következő módszerrel érdemes használni. -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.4. A vizuális szerkesztők használata</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">61. oldal</span></p> <p style="top:150.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Érdemes megemlíteni néhány funkciót, amit a szerző szándékosan nem szokott engedé-</span></p> <p style="top:163.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lyezni tartalomszerkesztők számára.</span></p> <p style="top:184.8pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">A weboldal egységes látványvilága miatt nem javasolt:</span></b></p> <p style="top:211.0pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">balra, középre és jobbra igazítás</span></p> <p style="top:231.3pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">térközök, behúzások</span></p> <p style="top:251.5pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">színes betűk és hátterek</span></p> <p style="top:271.8pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">betűtípusok</span></p> <p style="top:292.0pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kisebb és nagyobb betűk</span></p> <p style="top:312.3pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">bekezdések közötti elválasztó vonalak</span></p> <p style="top:332.5pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">képek balra, jobbra igazítása vegyesen</span></p> <p style="top:354.3pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Tipográfiai okok miatt nem javasolt:</span></b></p> <p style="top:380.5pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">sorkizárt igazítás </span></p> <p style="top:393.1pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(az elválasztás hiánya miatt)</span></p> <p style="top:413.3pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">aláhúzás </span></p> <p style="top:425.9pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(a linkekkel való összekeverhetőség, és az olvashatóság jelentős romlása miatt)</span></p> <p style="top:446.1pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">a csupa nagybetűs, szóközzel ritkított írásmód </span></p> <p style="top:458.7pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(ezek az írógépes korszak maradványai, erre jobb megoldás a dőlt, és félkövér for-</span></p> <p style="top:471.2pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">mázás konzekvens alkalmazása)</span></p> <p style="top:491.5pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">táblázatok </span></p> <p style="top:504.0pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(ami papíron kifér és olvasható, az egy weboldalon általában használhatatlan; </span></p> <p style="top:516.6pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kisebb táblázatokat pedig sokszor inkább felsorolásként érdemes</span><sup><span style="font-family:MagyarLinLibertine,serif;font-size:6.4pt;color:#000000">48</span></sup><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> közzétenni)</span></p> <p style="top:536.8pt;left:127.7pt;line-height:11.0pt"><span style="font-family:OpenSymbol,serif;font-size:11.0pt;color:#000000"></span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">képek körbefuttatása, speciális esetektől eltekintve </span></p> <p style="top:549.4pt;left:141.9pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">(általában sokkal kevesebb hely van a nyomtatott médiához képest)</span></p> <p style="top:577.4pt;left:113.6pt;line-height:16.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:16.0pt;color:#000000">2.4.2. Ajánlott módszer</span></b></p> <p style="top:609.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A vizuális szerkesztőket – a szerző véleménye szerint – a következő módszerrel érdemes </span></p> <p style="top:622.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">használni.</span></p> <p style="top:641.1pt;left:127.7pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">1.</span></p> <p style="top:641.1pt;left:145.7pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">írjuk meg helyben, vagy illesszük be a nyers szöveget</span></p> <p style="top:659.9pt;left:127.7pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">2.</span></p> <p style="top:659.9pt;left:145.7pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">állítsuk be a szöveg struktúráját (kevésbé szerencsés megfogalmazás a formázás)</span></p> <p style="top:678.6pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Először is nézzük meg, mit szeretnénk elérni, mi lesz a végeredmény (2.37. ábra).</span></p> <p style="top:708.8pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">48 A szerző e mondatok írása közben is először táblázatot akart alkalmazni, de inkább a felsorolások, </span></p> <p style="top:719.1pt;left:127.7pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">sortörések és zárójelek alkalmazása mellett döntött.</span></p> </div>	page_356.png	2.4. A vázuális szerkesztők használata 61. oldal Érdemes megemlíteni néhány funkciót, amit a szerző szándékosan nem szokott engedé- lyezni tartalomszerkesztők számáro. A weboldal egységes látványvilága miatt nem javasolt: balra, középre és jobbra igazi térközök, behúzások színes betük és hátterek betűtípusok kisebb és nagyobb betűk bekezdések közötti elválasztó vonalak képek balra, jobbra igazítása vegyesen Tipográfiai okok miatt nem javasolt; m sorkizárt igazítás (az elválasztás hiánya miatt) áhúzás (a inkekkel való összekeverhetőség, és az olvashatóság jelentős romlása miatt) M. a csupa nagybetűs, szóközzel ritkított írásmód (ezek az írógépes korszak maradványai, erre jobb megoldás a dölt, és félkövér for- mázás konzekvens alkalmazása) m táblázatok (ami papíron kifér és olvasható, az egy weboldalon általában használhot kásebb táblázatokat pedig sokszor inkább felsorolásként érdeme; M. képek közbefuttatása, spes dáltalában sokkal kevesebb. lis esetektől eltekintve tely van a nyomtatott médiához képest) 2.4.2. Ajánlott módszer A vizuális szerkesztőket - a szerző véleménye szerint - a következő módszerrel érdemes használni. 1. írjuk meg helyben, vagy illesszük be a nyers szöveget 2. állítsuk be a szöveg struktúráját (kevésbé szerencsés megfogalmazás a formázás) Először is nézzük meg, mit szeretnénk elérni, mi lesz a végeredmény (237. ábra). 18. A szerző e mondatok iisa közben és először tiblázotot ar alkalmazni,de nkább a felerolások,
2.4. A vizuális szerkesztők használata 63. oldal Látszik a két bekezdés bal felső sarkában a P (paragraph, vagyis bekezdés) betű. A Forráskód gombra kattintva meg is nézhetjük a háttérben készülő HTML szöveget (2.40. ábra). Ha szükséges, itt is belejavíthatunk, de bármikor visszatérhetünk a Forráskód gomb ismételt lenyomásával. Ezzel a módszerrel nem csak a leendő bekezdéseinket, hanem a felsorolásokat, címeket is érdemes először elkészíteni, és csak utána formázni. Szöveg beillesztése Példaként nézzük meg azt az esetet is, amikor a szöveg már kész van valahol (pl. Word dokumentumként), csak beilleszteni szeretnénk a szerkesztő felületbe. Először is vigyük a szövegkurzort a 2. bekezdés legvégére, és az Enter leütésével hozzunk létre egy új, üres bekezdést (2.41. ábra). 2.39. ábra. Szöveg bekezdésekre tördelése az Enter billentyűvel 2.40. ábra. Forráskód megtekintése	##### 2.4. A vizuális szerkesztők használata 63. oldal _2.39. ábra. Szöveg bekezdésekre tördelése az Enter billentyűvel_ Látszik a két bekezdés bal felső sarkában a P (paragraph, vagyis bekezdés) betű. A Forráskód gombra kattintva meg is nézhetjük a háttérben készülő HTML szöveget (2.40. ábra). _2.40. ábra. Forráskód megtekintése_ Ha szükséges, itt is belejavíthatunk, de bármikor visszatérhetünk a Forráskód gomb ismételt lenyomásával. Ezzel a módszerrel nem csak a leendő bekezdéseinket, hanem a felsorolásokat, címeket is érdemes először elkészíteni, és csak utána formázni. #### Szöveg beillesztése Példaként nézzük meg azt az esetet is, amikor a szöveg már kész van valahol (pl. Word dokumentumként), csak beilleszteni szeretnénk a szerkesztő felületbe. Először is vigyük a szövegkurzort a 2. bekezdés legvégére, és az Enter leütésével hozzunk létre egy új, üres bekezdést (2.41. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.4. A vizuális szerkesztők használata</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">63. oldal</span></p> <p style="top:341.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Látszik a két bekezdés bal felső sarkában a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">P</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> (paragraph, vagyis bekezdés) betű.</span></p> <p style="top:360.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Forráskód</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombra kattintva meg is nézhetjük a háttérben készülő HTML szöveget (2.40. </span></p> <p style="top:373.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ábra).</span></p> <p style="top:519.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ha szükséges, itt is belejavíthatunk, de bármikor visszatérhetünk a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Forráskód</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gomb ismé-</span></p> <p style="top:532.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">telt lenyomásával.</span></p> <p style="top:551.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Ezzel a módszerrel nem csak a leendő bekezdéseinket, hanem a felsorolásokat, címeket is </span></p> <p style="top:563.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">érdemes először elkészíteni, és csak utána formázni.</span></p> <p style="top:585.5pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Szöveg beillesztése</span></b></p> <p style="top:610.2pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Példaként nézzük meg azt az esetet is, amikor a szöveg már kész van valahol (pl. Word do-</span></p> <p style="top:622.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">kumentumként), csak beilleszteni szeretnénk a szerkesztő felületbe.</span></p> <p style="top:641.5pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Először is vigyük a szövegkurzort a 2. bekezdés legvégére, és az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Enter</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> leütésével hozzunk </span></p> <p style="top:654.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">létre egy új, üres bekezdést (2.41. ábra).</span></p> <p style="top:317.9pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.39. ábra. 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A vázuális szerkesztők használata 63. oldal T— . EZET IK FL TTT P36 BE zöbb szóvaljelezoezbeteen szgam $é a9. öktató, progzaczozó, de lepnkább azt tartoasfoctosaak. elmocdsat hogy Jézs Kroztos váltsága álal lten gyerzcke lebetek Szirooczcs ez a kezfootosatb sokdáz er találtara t a gyélekezetet ah mzaradóktalazal egyétt tudjok zzécni ter é egyétt todonkaz ezberek felé s szólslos a998-ben azonban megtaláta a Koskezzét Baptits Gyülekezetet, áholszolgáló testvérekze é otlhonzeletem v 9 4 239. ábra. Szöveg bekezdésekre tördelése az Enter billentyűvel Látszik a két bekezdés bal felső sarkában a P(paragraph, vagyis bekezdés) betű. A Forráskód gombra kattintva meg is nézhetjük a háttérben készülő HTML szöveget (240. ébr T— , E rondndó - 2 2t tezscn Zootzsnak cizondsni, pegy 7szús Kzisztes múzesága aztal lztez . myölykezetet; tc mszegértsissel ö9yott tedjok öizsérni Zszent, ós sgyost " jllgtlses oyaltkszetet, azol szotoáló keztvzekze é9 otrbeZa lelzez s 240. ábra. Forráskód megtekintése Ha szükséges, ítt is belejavíthatunk, de bármikor visszatérhetünk a Forráskód gomb ismé- telt lenyomásával. Ezzel a módszerrel nem csak a Ieendő bekezdéseinket, hanem a felsorolásokat, címeket is. érdemes először elkészíteni, és csak utána formázni. Szöveg beillesztése Példaként nézzük meg azt az esetet is, amikor a szöveg már kész van valahol (pl. Word do- kumentumként), csak bellleszteni szeretnénk a szerkesztő felületbe. Először is vigyük a szövegkurzort a 2. bekezdés legvégére, és az Enter leütésével hozzunk létre egy új, üres bekezdést (2.41. ábra).
2.4. A vizuális szerkesztők használata 65. oldal A Rendben előtt érdemes még arra figyelni, hogy a majdani bekezdések között pontosan egy üres sor legyen, mint az ábrán is. Ha ugyanis nincs üres sor, akkor ott a szerkesztő nem önálló bekezdést, hanem csak egy új sort fog kezdeni. Szövegstruktúra kialakítása Bár a pillanatnyi állapot is jól olvasható szöveget eredményez, érdemes néhány soron belüli kiemelést alkalmazni, valamint linkeket elhelyezni. Jelöljük ki az Isten gyermeke szöveget, majd kattintsunk a Félkövér gombon pont úgy, mintha a szövegszerkesztőnket használnánk (2.44. ábra). Hasonlóan emeljük ki a Családom szót is. Készítsünk néhány linket is a szövegbe. Hivatkozzunk az említett intézményekre. Példaként jelöljük ki a Kecskeméti Baptista Gyülekezetet szöveget, és kattintsunk a Hivatkozás beillesztése/módosítása gombra (2.45. ábra). 2.43. ábra. Beillesztés formázatlan szövegként 2.44. ábra. Félkövér formázás	##### 2.4. A vizuális szerkesztők használata 65. oldal _2.43. ábra. Beillesztés formázatlan szövegként_ A Rendben előtt érdemes még arra figyelni, hogy a majdani bekezdések között pontosan egy üres sor legyen, mint az ábrán is. Ha ugyanis nincs üres sor, akkor ott a szerkesztő nem önálló bekezdést, hanem csak egy új sort fog kezdeni. #### Szövegstruktúra kialakítása Bár a pillanatnyi állapot is jól olvasható szöveget eredményez, érdemes néhány soron belüli kiemelést alkalmazni, valamint linkeket elhelyezni. Jelöljük ki az Isten gyermeke szöveget, majd kattintsunk a Félkövér gombon pont úgy, mintha a szövegszerkesztőnket használnánk (2.44. ábra). _2.44. ábra. Félkövér formázás_ Hasonlóan emeljük ki a Családom szót is. Készítsünk néhány linket is a szövegbe. Hivatkozzunk az említett intézményekre. Példaként jelöljük ki a Kecskeméti Baptista Gyülekezetet szöveget, és kattintsunk a Hivatkozás _beillesztése/módosítása gombra (2.45. ábra)._ -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.4. A vizuális szerkesztők használata</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">65. oldal</span></p> <p style="top:383.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Rendben</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> előtt érdemes még arra figyelni, hogy a majdani bekezdések között pontosan </span></p> <p style="top:396.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">egy üres sor legyen, mint az ábrán is. Ha ugyanis nincs üres sor, akkor ott a szerkesztő </span></p> <p style="top:408.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">nem önálló bekezdést, hanem csak egy új sort fog kezdeni.</span></p> <p style="top:430.7pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Szövegstruktúra kialakítása</span></b></p> <p style="top:455.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Bár a pillanatnyi állapot is jól olvasható szöveget eredményez, érdemes néhány soron be-</span></p> <p style="top:468.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lüli kiemelést alkalmazni, valamint linkeket elhelyezni.</span></p> <p style="top:486.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Jelöljük ki az </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Isten gyermeke</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szöveget, majd kattintsunk a Félkövér gombon pont úgy, </span></p> <p style="top:499.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">mintha a szövegszerkesztőnket használnánk (2.44. ábra).</span></p> <p style="top:627.1pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Hasonlóan emeljük ki a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Családom</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szót is.</span></p> <p style="top:645.8pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Készítsünk néhány linket is a szövegbe. Hivatkozzunk az említett intézményekre. Példa-</span></p> <p style="top:658.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ként jelöljük ki a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Kecskeméti Baptista Gyülekezetet</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szöveget, és kattintsunk a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Hivatkozás </span></i></p> <p style="top:670.9pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">beillesztése/módosítása</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> gombra (2.45. ábra).</span></p> <p style="top:359.9pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.43. ábra. 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A vázuális szerkesztők használata 65. oldal 902. : Belkesztés formázatian szövegként . t ] O s zögene kenaten : T Ara] . kór tor Oa é: Áont s öl önyene Döta $r o a. . anioszós towesszotat Loci Czen 0r aa h 8200 a sgyér vitozés tráoan válok Oa síó ottojosztt 91 szitapcsozaos o] sss G , 243. ábra. Beillesztés formázatlan szövegként A Rendben előtt érdemes még arra figyelni, hogy a majdani bekezdések között pontosan. egy űres sor legyen, mint az ábrán is. Ha ugyanis nincs űres sor, akkor ott a szerkesztő mem önálló bekezdést, hanem csak egy új sort fog kezdeni. Szövegstruktúra kialakítása Bár a pillanatnyi állapot is jól olvasható szöveget eredményez, érdemes néhány soron be- tüli kiemelést alkalmazni, valamint limkeket elhelyezni. Jelöljük ki az Isten gyermeke szöveget, majd kattintsunk a Félkövér gombon pont úgy, mintha a szövegszerkesztönket használnánk (2.és. ábra). T— DEETEETET ELETLE S TET 36. - TI gélkövér lellezzbetem magar $ér a7a, cktató, progzaznozó, de Jegirkáb azt artocn fontocnak elmondani, hogy Józzs Keisztus váltsága általJEZZTZEEZT tebetok.Számorara ez a L dezfontosalb 244. ábra. Félkövér formázás Hasonlóan emeljük ki a Családom szótis. Készítsünk néhány linket is a szövegbe. Hivatkozzunk az említett intézményekre. Péld; ként jelöljük ki a Kecskeméti Baptista Gyűlekezetet szöveget, és kattintsunk a Hivatkozás beillesztése/módosítása gombra (2.45.
66. oldal 2. A Drupal felhasználói szemmel Külső weboldalra mutató link esetén (mint most is) elegendő a webcímet begépelni vagy beilleszteni a http:// nélkül (2.46. ábra). A Kecskeméti Főiskola és GAMF Karának szavakra ugyanígy elkészíthetjük a linkeket. Belső (a weboldalon belüli) link esetén érdemes egy másik ablakban/fülön megnyitni a célul kitűzött oldalt, és a domain név utáni részt, a / jellel kezdődően másoljuk a vágólapra. Példaként a vállalok szóra készítsünk egy linket. Az oldal a http://nagygusztav.hu/honlapfejlesztest-tanacsadast-vallalok címen érhető el, így a /honlap-fejlesztest-tanacsadast-vallalok szövegre lesz szükségünk. Ezt illesszük be a Hivatkozás tulajdonságai felugró ablak Hivatkozás mezőjébe (2.47. ábra). 2.45. ábra. Link létrehozása 2.46. ábra. Hivatkozás megadása	##### 66. oldal 2. A Drupal felhasználói szemmel _2.45. ábra. Link létrehozása_ Külső weboldalra mutató link esetén (mint most is) elegendő a webcímet begépelni vagy beilleszteni a http:// nélkül (2.46. ábra). _2.46. ábra. Hivatkozás megadása_ A Kecskeméti Főiskola és GAMF Karának szavakra ugyanígy elkészíthetjük a linkeket. Belső (a weboldalon belüli) link esetén érdemes egy másik ablakban/fülön megnyitni a célul kitűzött oldalt, és a domain név utáni részt, a / jellel kezdődően másoljuk a vágólapra. Példaként a vállalok szóra készítsünk egy linket. Az oldal a http://nagygusztav.hu/honlapfejlesztest-tanacsadast-vallalok címen érhető el, így a /honlap-fejlesztest-tanacsadast-vallalok szövegre lesz szükségünk. Ezt illesszük be a Hivatkozás tulajdonságai felugró ablak Hivat_kozás mezőjébe (2.47. ábra)._ -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:85.2pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">66. oldal</span></p> <p style="top:114.6pt;left:322.1pt;line-height:12.0pt"><span style="font-family:MagyarLinuxLibertineO,serif;font-size:12.0pt;color:#000000">2. A Drupal felhasználói szemmel</span></p> <p style="top:255.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Külső weboldalra mutató link esetén (mint most is) elegendő a webcímet begépelni vagy </span></p> <p style="top:268.2pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">beilleszteni a </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">http://</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> nélkül (2.46. ábra).</span></p> <p style="top:529.8pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Kecskeméti Főiskola</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> és </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">GAMF Karának</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szavakra ugyanígy elkészíthetjük a linkeket.</span></p> <p style="top:548.6pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Belső (a weboldalon belüli) link esetén érdemes egy másik ablakban/fülön megnyitni a cé-</span></p> <p style="top:561.1pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">lul kitűzött oldalt, és a domain név utáni részt, a / jellel kezdődően másoljuk a vágólapra.</span></p> <p style="top:579.9pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Példaként a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">vállalok</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> szóra készítsünk egy linket. Az oldal a </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">http://nagygusztav.hu/honlap-</span></p> <p style="top:593.2pt;left:85.2pt;line-height:10.0pt"><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">fejlesztest-tanacsadast-vallalok</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> címen érhető el, így a </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">/honlap-fejlesztest-tanacsadast-vallalok </span></p> <p style="top:605.0pt;left:85.2pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">szövegre lesz szükségünk. Ezt illesszük be a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Hivatkozás tulajdonságai</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> felugró ablak </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Hivat-</span></i></p> <p style="top:617.5pt;left:85.2pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">kozás</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> mezőjébe (2.47. ábra).</span></p> <p style="top:231.7pt;left:85.3pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.45. ábra. 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A Drupal felhasználói szemmel — x. GEE TTT FE 1 együrt todunkaz emberek feló is , Hivatkozás bedlesztése/módosítása ) a LZEE " ter ahel szsáló: 245. ábra. Link létrehozása Külső weboldalra mutató lnk esetén (mint most is) elegendő a webcimet begépelni vagy beillszteni a heto // nélkül (2.46. ábra). E v két Sank Dariés Átel, éskét 246. ábna. Hivatkozás megadása A Kecskeméti Főiskola és GAMF Karának szavakra ugyanígy elkészíthetjük a Tinkeket. Belső (a weboldalon belülí) link esetén érdemes egy másik ablakban/fülön megnyítni a cé. tul kítözőtt oldalt, és a domain név utáni részt, a / jellel kezdődően másoljuk a vágólapr Példaként a vállalok szóra készítsünk egy linket. Az oldal a Mto./nagygusztav.hu/honlap. fejlesztest.tanacsadast-vallalok címen érhető l, így a /honiap-fejlesztest:tanacsadast-vallalok szövegre lesz szükségünk. Ezt illesszük be a Hivatkozás tulajdonságai felugró ablak Hivat. kozás mezőjébe (247. ábra).
2.4. A vizuális szerkesztők használata 67. oldal Érdemes megfigyelni, hogy a kezdő / jel miatt a Protokoll a korábbi http:// helyett <más>ra váltott. Ez a helyes működés része. Ha esetleg nem történne meg automatikusan, a Protokollt kézzel érdemes így beállítani. A szövegbe ágyazott belső linkek használatának kockázata is van. Ha a hivatkozott oldal útvonala (útvonal álneve) megváltozik, akkor az így készített link „eltörik”. Érdemes megnézni a HTML forráskódot is, amit végül is elértünk a vizuális szerkesztő használatával, de akár kézzel is gépelhettük volna (2.48. ábra). A kitűzött célt (2.37. ábra) ezzel elértük, elmenthetjük a munkánkat. Felsorolás és számozás kialakítása Mivel nem magától értetődő, nézzünk egy példát a felsorolás, számozás kialakítására. Példánkban a http://nagygusztav.hu/online-tanfolyamok oldal másolatát készítjük el. A szöveg vágólapra másolása, és Beillesztés formázatlan szövegként (2.42. ábra) funkció használata után elég sok problémát mutat (2.49. ábra). 2.47. ábra. Belső link létrehozása 2.48. ábra. Az előállított HTML kód	##### 2.4. A vizuális szerkesztők használata 67. oldal _2.47. ábra. Belső link létrehozása_ Érdemes megfigyelni, hogy a kezdő / jel miatt a Protokoll a korábbi http:// helyett <más>ra váltott. Ez a helyes működés része. Ha esetleg nem történne meg automatikusan, a Pro_tokollt kézzel érdemes így beállítani._ A szövegbe ágyazott belső linkek használatának kockázata is van. Ha a hivatkozott oldal útvonala (útvonal álneve) megváltozik, akkor az így készített link „eltörik”. Érdemes megnézni a HTML forráskódot is, amit végül is elértünk a vizuális szerkesztő használatával, de akár kézzel is gépelhettük volna (2.48. ábra). _2.48. ábra. Az előállított HTML kód_ A kitűzött célt (2.37. ábra) ezzel elértük, elmenthetjük a munkánkat. #### Felsorolás és számozás kialakítása Mivel nem magától értetődő, nézzünk egy példát a felsorolás, számozás kialakítására. Példánkban a http://nagygusztav.hu/online-tanfolyamok oldal másolatát készítjük el. A szöveg vágólapra másolása, és Beillesztés formázatlan szövegként (2.42. ábra) funkció használata után elég sok problémát mutat (2.49. ábra). -----	<div id="page0" style="width:595.0pt;height:842.0pt"> <p style="top:114.6pt;left:113.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">2.4. A vizuális szerkesztők használata</span></p> <p style="top:114.6pt;left:470.6pt;line-height:12.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:12.0pt;color:#000000">67. oldal</span></p> <p style="top:286.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Érdemes megfigyelni, hogy a kezdő </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">/</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> jel miatt a Protokoll a korábbi </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">http://</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> helyett </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000"><más></span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">-</span></p> <p style="top:299.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">ra váltott. Ez a helyes működés része. Ha esetleg nem történne meg automatikusan, a </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Pro-</span></i></p> <p style="top:312.0pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">tokoll</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">t kézzel érdemes így beállítani.</span></p> <p style="top:330.5pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">A szövegbe ágyazott belső linkek használatának kockázata is van. Ha a hivatkozott oldal útvonala (útvonal ál-</span></p> <p style="top:340.8pt;left:113.6pt;line-height:9.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:9.0pt;color:#000000">neve) megváltozik, akkor az így készített link „eltörik”.</span></p> <p style="top:357.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Érdemes megnézni a HTML forráskódot is, amit végül is elértünk a vizuális szerkesztő </span></p> <p style="top:370.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">használatával, de akár kézzel is gépelhettük volna (2.48. ábra).</span></p> <p style="top:575.9pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A kitűzött célt (2.37. ábra) ezzel elértük, elmenthetjük a munkánkat.</span></p> <p style="top:597.7pt;left:113.6pt;line-height:13.0pt"><b><span style="font-family:MagyarLinLibertineB,serif;font-size:13.0pt;color:#000000">Felsorolás és számozás kialakítása</span></b></p> <p style="top:622.4pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">Mivel nem magától értetődő, nézzünk egy példát a felsorolás, számozás kialakítására. Pél-</span></p> <p style="top:635.0pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">dánkban a </span><span style="font-family:Calibri,serif;font-size:10.0pt;color:#4c4c4c">http://nagygusztav.hu/online-tanfolyamok</span><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> oldal másolatát készítjük el.</span></p> <p style="top:653.7pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">A szöveg vágólapra másolása, és </span><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">Beillesztés formázatlan szövegként</span></i><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000"> (2.42. ábra) funkció </span></p> <p style="top:666.3pt;left:113.6pt;line-height:11.0pt"><span style="font-family:MagyarLinLibertine,serif;font-size:11.0pt;color:#000000">használata után elég sok problémát mutat (2.49. ábra).</span></p> <p style="top:263.0pt;left:113.6pt;line-height:11.0pt"><i><span style="font-family:MagyarLinLibertineI,serif;font-size:11.0pt;color:#000000">2.47. ábra. 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