4y^{''}-y=e^{x/2}x

解答

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

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}

求解步骤

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

求解线性方程:

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}

\frac{d ^{2} y}{d x ^{2}} + y = 3 sin (2 x) + x cos (2 x)

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

y^{^{''}} - 3 y^{^{'}} + 5 = e^{2 x} - 3 cos (3 x)

(D^{2} + D) (y) = sin (x)

4 y^{^{''}} - y^{^{'}} + 6 = 0