Solve $x^2u_{xx}-u_{yy}= -2x$

Question

I'm trying to solve the following initial value problem

$x^2u_{xx}-u_{yy}= -2x, x>0, y>0$
$u(x,y) = 0,\,\, u_y(x,y) = 0$ on $\Gamma:=\{(x,y)\in\mathbb{R}^2\,:\,x>0,\,y = 0\}$

With the substitution $\xi = y-\log(x)$, $\eta = y+\log(x)$. From here, I need to solve

$-4u_{\xi\xi}-u_{\xi}-u_\eta = -2\exp\left(\frac{\eta-\xi}{2}\right)$

How can I proceed from this point? Is there some other approach for finding the exact solution to this problem?

Answer 1

$$x^2u_{xx}-u_{yy}= -2x$$ HINT : $$u(x,y)=v(x,y)-2(x\ln(x)-x)$$ $$x^2v_{xx}-v_{yy}= 0$$ Now one can try the separation of variables : $\quad v(x,y)=X(x)Y(y)$