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

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

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

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

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

Solution steps

y^{''} (x) - 2 y = e^{2 x} + x^{2} - 1

Solve linear ODE:

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

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

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