4y^{''}+4y^'+y=xe^{*x/2}sin(x)

解

4 y^{^{''}} + 4 y^{^{'}} + y = x e^{\cdot \frac{x}{2}} sin (x)

y = c_{1} e^{- \frac{x}{2}} + c_{2} e^{- \frac{x}{2}} x + \frac{- e ^{\frac{x}{2}} x cos ( x ) + e ^{\frac{x}{2}} sin ( x ) + e ^{\frac{x}{2}} cos ( x )}{8}

解答ステップ

4 y^{''} (x) + 4 y^{'} (x) + y = x e^{\cdot \frac{x}{2}} sin (x)

線形 ODE を解く:

y = c_{1} e^{- \frac{x}{2}} + c_{2} e^{- \frac{x}{2}} x + \frac{- e ^{\frac{x}{2}} x cos ( x ) + e ^{\frac{x}{2}} sin ( x ) + e ^{\frac{x}{2}} cos ( x )}{8}

y = c_{1} e^{- \frac{x}{2}} + c_{2} e^{- \frac{x}{2}} x + \frac{- e ^{\frac{x}{2}} x cos ( x ) + e ^{\frac{x}{2}} sin ( x ) + e ^{\frac{x}{2}} cos ( x )}{8}