What is the solution for y^{''}+2y^'+y=3x*e^{-x} ?

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

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y^{''}+2y^'+y=3x*e^{-x}

y^{^{''}} + 2 y^{^{'}} + y = 3 x \cdot e^{- x}

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

Solution steps

y^{''} (x) + 2 y^{'} (x) + y = 3 x \cdot e^{- x}

Solve linear ODE:

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

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

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