What is the solution for (d^2y)/(dt^2)-y=e^t ?

The solution for (d^2y)/(dt^2)-y=e^t is y=c_{1}e^t+c_{2}e^{-t}+(e^tt)/2

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(d^2y)/(dt^2)-y=e^t

\frac{d ^{2} y}{d t ^{2}} - y = e^{t}

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

Solution steps

\frac{d ^{2} y}{d t ^{2}} - y = e^{t}

Solve linear ODE:

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

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

x^{^{''}} - 5 x^{^{'}} + 4 x = 8 e^{t}

y^{^{''}} + 9 y = 10 e^{- t}

y^{^{''}} - 7 y^{^{'}} + 6 y = (x - 2) e^{x}

y^{^{''}} - 2 y^{^{'}} + 37 y = e^{x} cos (6 x)

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

What is the solution for (d^2y)/(dt^2)-y=e^t ?
The solution for (d^2y)/(dt^2)-y=e^t is y=c_{1}e^t+c_{2}e^{-t}+(e^tt)/2