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

The solution for y^{''}-2y+y=x^2e^x is y=c_{1}e^x+c_{2}e^{-x}+(e^xx^3)/6-(e^xx^2)/4+(e^xx)/4

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

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

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{x} x ^{3}}{6} - \frac{e ^{x} x ^{2}}{4} + \frac{e ^{x} x}{4}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{x} x ^{3}}{6} - \frac{e ^{x} x ^{2}}{4} + \frac{e ^{x} x}{4}

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{x} x ^{3}}{6} - \frac{e ^{x} x ^{2}}{4} + \frac{e ^{x} x}{4}

What is the solution for y^{''}-2y+y=x^2e^x ?
The solution for y^{''}-2y+y=x^2e^x is y=c_{1}e^x+c_{2}e^{-x}+(e^xx^3)/6-(e^xx^2)/4+(e^xx)/4