What is the solution for 2y^{''}+5y^'+2y=xe^{-x} ?

The solution for 2y^{''}+5y^'+2y=xe^{-x} is y=c_{1}e^{-x/2}+c_{2}e^{-2x}-e^{-x}x-e^{-x}

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2y^{''}+5y^'+2y=xe^{-x}

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

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

Solution steps

2 y^{''} (x) + 5 y^{'} (x) + 2 y = x e^{- x}

Solve linear ODE:

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

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

y^{^{''}} - 4 y^{^{'}} + 4 y = x^{2} \cdot e^{2 x} - e^{2 x}

y^{^{''}} + 0.7 y + 4 = cos (3 t)

y^{^{''}} - 4 y^{^{'}} + 3 y = e^{2 t}

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

1 y^{^{''}} + 5 y^{^{'}} + \frac{1}{\frac{1}{6}} y = 2 e^{2 t}

What is the solution for 2y^{''}+5y^'+2y=xe^{-x} ?
The solution for 2y^{''}+5y^'+2y=xe^{-x} is y=c_{1}e^{-x/2}+c_{2}e^{-2x}-e^{-x}x-e^{-x}