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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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