y^{''}-4y^'+5y=x^2e^3x

Lösung

y^{^{''}} - 4 y^{^{'}} + 5 y = x^{2} e^{3} x

y = e^{2 x} (c_{1} cos (x) + c_{2} sin (x)) + \frac{e ^{3} x ^{3}}{5} + \frac{12 e ^{3} x ^{2}}{25} + \frac{66 e ^{3} x}{125} + \frac{144 e ^{3}}{625}

Schritte zur Lösung

y^{''} (x) - 4 y^{'} (x) + 5 y = x^{2} e^{3} x

Solve linear ODE:

y = e^{2 x} (c_{1} cos (x) + c_{2} sin (x)) + \frac{e ^{3} x ^{3}}{5} + \frac{12 e ^{3} x ^{2}}{25} + \frac{66 e ^{3} x}{125} + \frac{144 e ^{3}}{625}

y = e^{2 x} (c_{1} cos (x) + c_{2} sin (x)) + \frac{e ^{3} x ^{3}}{5} + \frac{12 e ^{3} x ^{2}}{25} + \frac{66 e ^{3} x}{125} + \frac{144 e ^{3}}{625}

(d^{2} - 1) y = x

(d^{2} + 2) y = x^{2} e^{3} x

y^{^{''}} - 4 y^{^{'}} + 4 y = (x + 1) e^{2} x

(d^{2} - 4 d - 12) \cdot y = (x - 1) \cdot e^{- 2 x}

(d^{2} + 4 d + 4) y = x^{2} + e^{x} + cos (x)