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

\frac{d}{d x} (1 - e^{- 5 x})

x \frac{d y}{d x} = y^{2} - 4

x y^{^{'}} + y = 5 x y^{2}, y (1) = 3

\frac{( x - 1 ) ^{3}}{x ^{2}}

\int \frac{4}{4 + e ^{x}} d 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