What is the solution for y^{''}+6y^'+9y=2e^{-x} ?

The solution for y^{''}+6y^'+9y=2e^{-x} is y=c_{1}e^{-3x}+c_{2}xe^{-3x}+1/2 e^{-x}

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y^{''}+6y^'+9y=2e^{-x}

y^{^{''}} + 6 y^{^{'}} + 9 y = 2 e^{- x}

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

Solution steps

y^{''} (x) + 6 y^{'} (x) + 9 y = 2 e^{- x}

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} - 3 y = 8 e^{x} - 6

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

y^{^{'''}} - 2 y^{^{''}} - y^{^{'}} + 2 y = e^{7 t}

y^{^{''}} + 3 y^{^{'}} - 18 y = - 66 t + 36 t^{2}

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

What is the solution for y^{''}+6y^'+9y=2e^{-x} ?
The solution for y^{''}+6y^'+9y=2e^{-x} is y=c_{1}e^{-3x}+c_{2}xe^{-3x}+1/2 e^{-x}