What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2e^x ?

The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2e^x is y=c_{1}e^x+c_{2}xe^x+(e^xx^4)/(12)

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(d^2y)/(dx^2)-2(dy)/(dx)+y=x^2e^x

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

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{4}}{12}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{4}}{12}

y = c_{1} e^{x} + c_{2} x e^{x} + \frac{e ^{x} x ^{4}}{12}

What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2e^x ?
The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2e^x is y=c_{1}e^x+c_{2}xe^x+(e^xx^4)/(12)