What is the solution for y^{''}+7y^'+12y=e^t,y(0)=0,y^'(0)=1 ?

The solution for y^{''}+7y^'+12y=e^t,y(0)=0,y^'(0)=1 is y=(4/5-1/20)e^{-3t}-4/5 e^{-4t}+1/20 e^t

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y^{''}+7y^'+12y=e^t,y(0)=0,y^'(0)=1

y^{^{''}} + 7 y^{^{'}} + 12 y = e^{t}, y (0) = 0, y^{^{'}} (0) = 1

y = (\frac{4}{5} - \frac{1}{20}) e^{- 3 t} - \frac{4}{5} e^{- 4 t} + \frac{1}{20} e^{t}

Solution steps

y^{''} (t) + 7 y^{'} (t) + 12 y = e^{t}

Solve linear ODE:

y = (\frac{4}{5} - \frac{1}{20}) e^{- 3 t} - \frac{4}{5} e^{- 4 t} + \frac{1}{20} e^{t}

y = (\frac{4}{5} - \frac{1}{20}) e^{- 3 t} - \frac{4}{5} e^{- 4 t} + \frac{1}{20} e^{t}

What is the solution for y^{''}+7y^'+12y=e^t,y(0)=0,y^'(0)=1 ?
The solution for y^{''}+7y^'+12y=e^t,y(0)=0,y^'(0)=1 is y=(4/5-1/20)e^{-3t}-4/5 e^{-4t}+1/20 e^t