What is the solution for y^{''}-y^'+0.25y=3+e^{x/2} ?

The solution for y^{''}-y^'+0.25y=3+e^{x/2} is y=c_{1}e^{x/2}+c_{2}e^{x/2}x+12+(e^{x/2}x^2)/2

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y^{''}-y^'+0.25y=3+e^{x/2}

y^{^{''}} - y^{^{'}} + 0.25 y = 3 + e^{\frac{x}{2}}

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{\frac{x}{2}} x + 12 + \frac{e ^{\frac{x}{2}} x ^{2}}{2}

Solution steps

y^{''} (x) - y^{'} (x) + 0.25 y = 3 + e^{\frac{x}{2}}

Solve linear ODE:

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{\frac{x}{2}} x + 12 + \frac{e ^{\frac{x}{2}} x ^{2}}{2}

y = c_{1} e^{\frac{x}{2}} + c_{2} e^{\frac{x}{2}} x + 12 + \frac{e ^{\frac{x}{2}} x ^{2}}{2}

y^{^{''}} + 2 y^{^{'}} - 3 y = 8 x e^{x}

y^{^{''}} + 2 y = 5 t^{2} - 3, y (0) = 0, y^{^{'}} (0) = - 6

10 y^{^{''}} + 1000 y = 140 cos (8 t), y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 9 y = 2 t^{2} - 4, y (0) = 0, y^{^{'}} (0) = - 4

y^{^{''}} + 3 y^{^{'}} + 4 y = 2 sin (3 t)

What is the solution for y^{''}-y^'+0.25y=3+e^{x/2} ?
The solution for y^{''}-y^'+0.25y=3+e^{x/2} is y=c_{1}e^{x/2}+c_{2}e^{x/2}x+12+(e^{x/2}x^2)/2