i need to solve

$\displaystyle x^2y''-xy'+y=4x^2$

by saying that $\displaystyle x=e^t$

and Y(t)=y(x(t))

then i need to find from them expresions for \fracxdy/dx and$\displaystyle x^2d^2y/dx^2$ which are expresions of dY/dt similar type

i tried like this:

$\displaystyle x=e^t$

$\displaystyle \frac{dx}{dt}=e^t$

Y(t)=y(x(t))

$\displaystyle \frac{dY(t)}{dt}=\frac{dy(x)}{dx}\frac{dx(t)}{dt}$

$\displaystyle \frac{dY(t)}{dt}=\frac{dy(x)}{dx}x$

$\displaystyle \frac{dY^2(t)}{dt}=\frac{d^2y(x)}{dx^2}x^2+\frac{d y(x)}{dx}$

$\displaystyle y'x=\frac{dY(t)}{dt}$

$\displaystyle y''x^2=\frac{dY^2(t)}{dt}-y'$

so

$\displaystyle \frac{dY^2(t)}{dt}-y'-\frac{dY(t)}{dt}+y=4x^2$

what now

?