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

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

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

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

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{1}{9} e^{- 2 x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{1}{9} e^{- 2 x}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{1}{9} e^{- 2 x}

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