Cauchy-Euler equation of differential equation

Question

Solve the following equation: $$(x^2D^2 +4xD+2)y=e^x$$

I proceeded by using Cauchy-Euler equation but couldn't solve it.

Answer 1

Hint

Let $y=\frac z {x^2}$. This would give a very simple equation

Answer 2

You can proceed this way: $$(x^2D^2 +4xD+2)y=e^x$$ $$x^2y''+2xy'+2xy'+2y=e^x$$ $$(x^2y')'+(2xy)'=e^x$$ $$(x^2y'+2xy)'=e^x$$ $$(x^2y)''=e^x$$ Integrate twice.