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

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

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

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} - e^{x} ln (x) - e^{x}

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

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