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

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

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y^{''}+3y^'+2y=xe^{-x}

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y^{^{'}} + 3 y = x^{2} - 2

3 y^{^{''}} - 5 y^{^{'}} + 2 y = 6 x e^{- 2 x}

y^{^{''}} + 2 y^{^{'}} + y = sin (x) + 7 cos (2 x)

y^{^{''}} + 4 y^{^{'}} + 4 y = (7 + x) e - 2 x

y^{^{''}} - y^{^{'}} - 6 y = e^{t}

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