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

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

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y^{''}+2y^'+y= 1/x e^{-x}

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

y = c_{1} e^{- x} + c_{2} x e^{- x} - e^{- x} x + e^{- x} x ln (x)

Solution steps

y^{''} (x) + 2 y^{'} (x) + y = \frac{1}{x} e^{- x}

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} - e^{- x} x + e^{- x} x ln (x)

y = c_{1} e^{- x} + c_{2} x e^{- x} - e^{- x} x + e^{- x} x ln (x)

y^{^{''}} + x + 4 y = 0

y^{^{''}} + 6 y^{^{'}} + 9 y = x e^{4 x}

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

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

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

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