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

Lösung

y^{^{''}} - 2 y^{^{'}} + y = e^{x} \frac{1}{x ^{2} + 1}

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

Schritte zur Lösung

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

Solve linear ODE:

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

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

y^{^{''}} - 6 y^{^{'}} + 9 y = e^{3 x} ln (x^{2} + 1)

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{2} e^{3}

y^{^{''}} - 2 y^{^{'}} + y = 2 x + x^{2} + sin (3 x)

y^{^{''}} + 4 y = cos (2 t), y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + y^{^{'}} + 5 y = 4 e^{- t} cos (2 t)