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

The solution for y^{''}+y^'=x^2e^x is y=c_{1}+c_{2}e^{-x}+(e^xx^2)/2-(3e^xx)/2+7/4 e^x

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

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

y = c_{1} + c_{2} e^{- x} + \frac{e ^{x} x ^{2}}{2} - \frac{3 e ^{x} x}{2} + \frac{7}{4} e^{x}

Solution steps

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

Solve linear ODE:

y = c_{1} + c_{2} e^{- x} + \frac{e ^{x} x ^{2}}{2} - \frac{3 e ^{x} x}{2} + \frac{7}{4} e^{x}

y = c_{1} + c_{2} e^{- x} + \frac{e ^{x} x ^{2}}{2} - \frac{3 e ^{x} x}{2} + \frac{7}{4} e^{x}

What is the solution for y^{''}+y^'=x^2e^x ?
The solution for y^{''}+y^'=x^2e^x is y=c_{1}+c_{2}e^{-x}+(e^xx^2)/2-(3e^xx)/2+7/4 e^x