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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y^{^{'}} + 5 y = tan (x)

x^{^{''}} + x = t + t^{2}

(D^{2} + 5 D + 6) y = e^{x}

\frac{d ^{2} y}{d t ^{2}} - 5 \frac{d y}{d t} + 5 y = 1 + t

y^{^{''}} + 4 y^{^{'}} + 20 y = 4 cos (2 x)

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