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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - y^{^{'}} - 2 y = 4 t^{2}, y (0) = 1, y^{^{'}} (0) = 4

y^{^{''}} + 25 y = 2 csc^{2} (5 t)

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = \frac{e ^{x}}{x ^{5}}

y^{^{''}} + 6 y^{^{'}} + 9 y = t

y^{^{''}} - y = 5 e^{4 t} + 6 t e^{4 t} + 3 t^{2} e^{4 t}

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