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

The solution for y^{''}+2y^'+y=x^2e^{-x} is y=c_{1}e^{-x}+c_{2}xe^{-x}+(e^{-x}x^4)/(12)

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

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

y = c_{1} e^{- x} + c_{2} x e^{- x} + \frac{e ^{- x} x ^{4}}{12}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} + \frac{e ^{- x} x ^{4}}{12}

y = c_{1} e^{- x} + c_{2} x e^{- x} + \frac{e ^{- x} x ^{4}}{12}

What is the solution for y^{''}+2y^'+y=x^2e^{-x} ?
The solution for y^{''}+2y^'+y=x^2e^{-x} is y=c_{1}e^{-x}+c_{2}xe^{-x}+(e^{-x}x^4)/(12)