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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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