What is the solution for x^{''}-x=e^{-t} ?

The solution for x^{''}-x=e^{-t} is x=c_{1}e^t+c_{2}e^{-t}-(e^{-t}t)/2

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x^{''}-x=e^{-t}

x^{^{''}} - x = e^{- t}

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

Solution steps

x^{''} (t) - x = e^{- t}

Solve linear ODE:

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

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

What is the solution for x^{''}-x=e^{-t} ?
The solution for x^{''}-x=e^{-t} is x=c_{1}e^t+c_{2}e^{-t}-(e^{-t}t)/2