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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 4 y = 6 cos (3 t + 3 0^{\circ})

y^{^{''}} - 7 y^{^{'}} + 6 y = 3 e^{t}

y^{^{''}} + 4 y = t e^{3 t} + 3 sin (4 t)

y^{^{''}} + 2 y^{^{'}} - 3 y = e^{x}

y^{^{''}} + 4 y^{^{'}} + 4 y = x^{2} - 2 x

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