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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 3 y^{^{'}} - 4 y = 4 x

y^{^{''}} - 2 y^{^{'}} + 5 y = - 8, y (0) = 2, y^{^{'}} (0) = 2

\frac{1}{2} y^{^{''}} + 2 y = 2 tan (2 t) - \frac{1}{2} e^{t}

y^{^{''}} - y = x^{2} e^{x} + 6

y^{^{''}} - y = x^{2} e^{x} + 9

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