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

解

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

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

解答ステップ

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

線形 ODE を解く:

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

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