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

The solution for y^{''}+y^'=e^{2t}+3te^t is y=c_{1}+c_{2}e^{-t}+1/6 e^{2t}+(3e^tt)/2-9/4 e^t

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

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

y = c_{1} + c_{2} e^{- t} + \frac{1}{6} e^{2 t} + \frac{3 e ^{t} t}{2} - \frac{9}{4} e^{t}

Solution steps

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

Solve linear ODE:

y = c_{1} + c_{2} e^{- t} + \frac{1}{6} e^{2 t} + \frac{3 e ^{t} t}{2} - \frac{9}{4} e^{t}

y = c_{1} + c_{2} e^{- t} + \frac{1}{6} e^{2 t} + \frac{3 e ^{t} t}{2} - \frac{9}{4} e^{t}

y^{^{''}} - 2 y^{^{'}} + 2 y = e^{x}

y^{^{''}} - 2 y^{^{'}} + y = 2 x

y^{^{''}} + 2 y^{^{'}} + 2 y = 5 e^{6 x}

\frac{d ^{2} x}{d t ^{2}} + x k + a = 0

y^{^{''}} - 2 y^{^{'}} + y = x^{\frac{1}{2}} e^{x}

What is the solution for y^{''}+y^'=e^{2t}+3te^t ?
The solution for y^{''}+y^'=e^{2t}+3te^t is y=c_{1}+c_{2}e^{-t}+1/6 e^{2t}+(3e^tt)/2-9/4 e^t