y^{''}-10y^'+25y=e^{5x}ln(4+x^2)

Lösung

y^{^{''}} - 10 y^{^{'}} + 25 y = e^{5 x} ln (4 + x^{2})

y = c_{1} e^{5 x} + c_{2} x e^{5 x} + \frac{1}{2} e^{5 x} x^{2} ln (x^{2} + 4) - \frac{3}{2} e^{5 x} x^{2} + 4 e^{5 x} x arctan (\frac{x}{2}) - 2 e^{5 x} ln (x^{2} + 4) + 2 e^{5 x}

Schritte zur Lösung

y^{''} (x) - 10 y^{'} (x) + 25 y = e^{5 x} ln (4 + x^{2})

Solve linear ODE:

y = c_{1} e^{5 x} + c_{2} x e^{5 x} + \frac{1}{2} e^{5 x} x^{2} ln (x^{2} + 4) - \frac{3}{2} e^{5 x} x^{2} + 4 e^{5 x} x arctan (\frac{x}{2}) - 2 e^{5 x} ln (x^{2} + 4) + 2 e^{5 x}

y = c_{1} e^{5 x} + c_{2} x e^{5 x} + \frac{1}{2} e^{5 x} x^{2} ln (x^{2} + 4) - \frac{3}{2} e^{5 x} x^{2} + 4 e^{5 x} x arctan (\frac{x}{2}) - 2 e^{5 x} ln (x^{2} + 4) + 2 e^{5 x}

y^{^{''}} - 2 y^{^{'}} - 3 y = 4 e^{x} - 9

y^{^{''}} - y^{^{'}} = e^{x}

y^{^{''}} + 4 y^{^{'}} + 4 y = 8 e^{- 2 x}

\frac{d ^{2} y}{d x ^{2}} - 8 \frac{d y}{d x} + 16 y = e^{4 x}

y^{^{''}} + y = 5 x sin (x)