What is the solution for y^{''}+3y^'+2y=6e^{-t},y(0)=1,y^'(0)=2 ?

The solution for y^{''}+3y^'+2y=6e^{-t},y(0)=1,y^'(0)=2 is y=-2e^{-t}+3e^{-2t}+6e^{-t}t

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y^{''}+3y^'+2y=6e^{-t},y(0)=1,y^'(0)=2

y^{^{''}} + 3 y^{^{'}} + 2 y = 6 e^{- t}, y (0) = 1, y^{^{'}} (0) = 2

y = - 2 e^{- t} + 3 e^{- 2 t} + 6 e^{- t} t

Solution steps

y^{''} (t) + 3 y^{'} (t) + 2 y = 6 e^{- t}

Solve linear ODE:

y = - 2 e^{- t} + 3 e^{- 2 t} + 6 e^{- t} t

y = - 2 e^{- t} + 3 e^{- 2 t} + 6 e^{- t} t

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

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} + 5 = 0

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

y^{^{''}} + 2 y^{^{'}} + 2 y = 6

y^{^{'''}} - 3 y^{^{''}} + 2 y^{^{'}} = x^{2} + 4 x + 8

What is the solution for y^{''}+3y^'+2y=6e^{-t},y(0)=1,y^'(0)=2 ?
The solution for y^{''}+3y^'+2y=6e^{-t},y(0)=1,y^'(0)=2 is y=-2e^{-t}+3e^{-2t}+6e^{-t}t