y^{''}+4y^'+4y=x^2e^{2x}

Lösung

y^{^{''}} + 4 y^{^{'}} + 4 y = x^{2} e^{2 x}

y = c_{1} e^{- 2 x} + c_{2} x e^{- 2 x} + \frac{e ^{2 x} x ^{2}}{16} - \frac{e ^{2 x} x}{16} + \frac{3}{128} e^{2 x}

Schritte zur Lösung

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

Solve linear ODE:

y = c_{1} e^{- 2 x} + c_{2} x e^{- 2 x} + \frac{e ^{2 x} x ^{2}}{16} - \frac{e ^{2 x} x}{16} + \frac{3}{128} e^{2 x}

y = c_{1} e^{- 2 x} + c_{2} x e^{- 2 x} + \frac{e ^{2 x} x ^{2}}{16} - \frac{e ^{2 x} x}{16} + \frac{3}{128} e^{2 x}

y^{^{''}} + 2 y^{^{'}} + 4 y = - 10 x e^{x}

y^{^{''}} + 4 y^{^{'}} + 5 = 50 x + 13 e^{3 x}

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} = 12 x

y^{^{''}} + 3 y^{^{'}} = 3 x

y^{^{''}} - y^{^{'}} - 2 y = - 4 x^{2} - 6 x + 1