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

The solution for y^{''}-4y=x+e^x is y=c_{1}e^{2x}+c_{2}e^{-2x}-x/4-1/3 e^x

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

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

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{x}{4} - \frac{1}{3} e^{x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{x}{4} - \frac{1}{3} e^{x}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{x}{4} - \frac{1}{3} e^{x}

y^{^{''}} + 6 y^{^{'}} + 8 y = x^{2}

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 2 y = e^{4 x}

y^{^{''}} - 4 y^{^{'}} - 3 y = cos (2 x)

y^{^{''}} + 4 y^{^{'}} = - 32 (x + 2), y (0) = 0, y^{^{'}} (0) = - 10

3 y^{^{''}} - 4 y^{^{'}} - 5 y = (x^{2} + 1) e^{- x}

What is the solution for y^{''}-4y=x+e^x ?
The solution for y^{''}-4y=x+e^x is y=c_{1}e^{2x}+c_{2}e^{-2x}-x/4-1/3 e^x