What is the solution for x^{''}+2x^'+5x=e^t ?

The solution for x^{''}+2x^'+5x=e^t is x=e^{-t}(c_{1}cos(2t)+c_{2}sin(2t))+1/8 e^t

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x^{''}+2x^'+5x=e^t

x^{^{''}} + 2 x^{^{'}} + 5 x = e^{t}

x = e^{- t} (c_{1} cos (2 t) + c_{2} sin (2 t)) + \frac{1}{8} e^{t}

Solution steps

x^{''} (t) + 2 x^{'} (t) + 5 x = e^{t}

Solve linear ODE:

x = e^{- t} (c_{1} cos (2 t) + c_{2} sin (2 t)) + \frac{1}{8} e^{t}

x = e^{- t} (c_{1} cos (2 t) + c_{2} sin (2 t)) + \frac{1}{8} e^{t}

What is the solution for x^{''}+2x^'+5x=e^t ?
The solution for x^{''}+2x^'+5x=e^t is x=e^{-t}(c_{1}cos(2t)+c_{2}sin(2t))+1/8 e^t