4(d^2y)/(dx^2)-y=xe^{x/2}

解

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

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{- \frac{x}{2}} + \frac{e ^{\frac{x}{2}} x ^{2}}{8} + \frac{- e ^{\frac{x}{2}} x + e ^{\frac{x}{2}}}{4}

解答ステップ

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

線形 ODE を解く:

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{- \frac{x}{2}} + \frac{e ^{\frac{x}{2}} x ^{2}}{8} + \frac{- e ^{\frac{x}{2}} x + e ^{\frac{x}{2}}}{4}

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{- \frac{x}{2}} + \frac{e ^{\frac{x}{2}} x ^{2}}{8} + \frac{- e ^{\frac{x}{2}} x + e ^{\frac{x}{2}}}{4}