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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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