(d^2y)/(dx^2)-2(dy)/(dx)=(e^x)/(x^2+1)

Lösung

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

y = c_{1} e^{2 x} + c_{2} + e^{2 x} \cdot \int \frac{1}{2 e ^{x} ( x ^{2} + 1 )} d x + \int - \frac{e ^{x}}{2 ( x ^{2} + 1 )} d x

Schritte zur Lösung

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} + e^{2 x} \cdot \int \frac{1}{2 e ^{x} ( x ^{2} + 1 )} d x + \int - \frac{e ^{x}}{2 ( x ^{2} + 1 )} d x

y = c_{1} e^{2 x} + c_{2} + e^{2 x} \cdot \int \frac{1}{2 e ^{x} ( x ^{2} + 1 )} d x + \int - \frac{e ^{x}}{2 ( x ^{2} + 1 )} d x

\frac{d ^{2} y}{d x ^{2}} - 3 \frac{d y}{d x} + 2 y = e^{3 x}

y^{^{''}} + 3 y^{^{'}} - 40 y = - 132 t + 80 t^{2}

y^{^{''}} + 8 y^{^{'}} + 12 = 0

y^{^{''}} + 2 y^{^{'}} + 2 y = 5 e^{t} + 2 t + 2

y^{^{'''}} = 3 y^{^{''}} + 9 e^{3} x + 5 cos (x)