What is the solution for y^{''}+y^'=e^{2t}+3+e^t ?

The solution for y^{''}+y^'=e^{2t}+3+e^t is y=c_{1}+c_{2}e^{-t}+1/6 e^{2t}+3t+1/2 e^t

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y^{''}+y^'=e^{2t}+3+e^t

y^{^{''}} + y^{^{'}} = e^{2 t} + 3 + e^{t}

y = c_{1} + c_{2} e^{- t} + \frac{1}{6} e^{2 t} + 3 t + \frac{1}{2} e^{t}

Solution steps

y^{''} (t) + y^{'} (t) = e^{2 t} + 3 + e^{t}

Solve linear ODE:

y = c_{1} + c_{2} e^{- t} + \frac{1}{6} e^{2 t} + 3 t + \frac{1}{2} e^{t}

y = c_{1} + c_{2} e^{- t} + \frac{1}{6} e^{2 t} + 3 t + \frac{1}{2} e^{t}

What is the solution for y^{''}+y^'=e^{2t}+3+e^t ?
The solution for y^{''}+y^'=e^{2t}+3+e^t is y=c_{1}+c_{2}e^{-t}+1/6 e^{2t}+3t+1/2 e^t