What is the solution for 4y^{''}-y=e^{x/2} ?

The solution for 4y^{''}-y=e^{x/2} is y=c_{1}e^{x/2}+c_{2}e^{-x/2}+(e^{x/2}x)/4

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{'''}} + 9 y^{^{''}} + 27 y^{^{'}} + 27 y = 5 e^{- 3 x}

(D^{2} - 5 D + 6) y = 3 e^{3 x} + e^{2 x}

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

6 y^{^{'''}} + 4 y^{^{'}} = e^{x} cos (2 x)

4 y^{^{''}} + 25 y = 41 e^{2 x}

What is the solution for 4y^{''}-y=e^{x/2} ?
The solution for 4y^{''}-y=e^{x/2} is y=c_{1}e^{x/2}+c_{2}e^{-x/2}+(e^{x/2}x)/4