What is the solution for y^{''}+9y^'+18y=x+e^{-x} ?

The solution for y^{''}+9y^'+18y=x+e^{-x} is y=c_{1}e^{-3x}+c_{2}e^{-6x}+x/(18)-1/36+1/10 e^{-x}

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y^{''}+9y^'+18y=x+e^{-x}

y^{^{''}} + 9 y^{^{'}} + 18 y = x + e^{- x}

y = c_{1} e^{- 3 x} + c_{2} e^{- 6 x} + \frac{x}{18} - \frac{1}{36} + \frac{1}{10} e^{- x}

Solution steps

y^{''} (x) + 9 y^{'} (x) + 18 y = x + e^{- x}

Solve linear ODE:

y = c_{1} e^{- 3 x} + c_{2} e^{- 6 x} + \frac{x}{18} - \frac{1}{36} + \frac{1}{10} e^{- x}

y = c_{1} e^{- 3 x} + c_{2} e^{- 6 x} + \frac{x}{18} - \frac{1}{36} + \frac{1}{10} e^{- x}

What is the solution for y^{''}+9y^'+18y=x+e^{-x} ?
The solution for y^{''}+9y^'+18y=x+e^{-x} is y=c_{1}e^{-3x}+c_{2}e^{-6x}+x/(18)-1/36+1/10 e^{-x}