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

The solution for y^{''}+2y^'+y=(e^{-x})/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=(e^{-x})/x

y^{^{''}} + 2 y^{^{'}} + y = \frac{e ^{- x}}{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{e ^{- x}}{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^{^{''}} - 4 y^{^{'}} = 1, y (0) = 1, y^{^{'}} (0) = 4

y^{^{''}} + 4 y = 4 sec (2 x)

y^{^{''}} - 2 y^{^{'}} + 2 y = 2 x - 2, y (π) = π, y (0) = 0

\frac{d ^{2} y}{d x ^{2}} + y = x^{2}

y^{^{''}} + y = - 16 cos (3 x)

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