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

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

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

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

x = c_{1} cos (t) + c_{2} sin (t) - \frac{1}{2} e^{t}

Solution steps

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

Solve linear ODE:

x = c_{1} cos (t) + c_{2} sin (t) - \frac{1}{2} e^{t}

x = c_{1} cos (t) + c_{2} sin (t) - \frac{1}{2} e^{t}

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