How to solve $4y^2u_{xx}+2(1-y^2)u_{xy}-\frac{2y}{1+y^2}(2u_x-u_y)=0$, $x\in\Bbb R, y>0$, $u|_{y=0}=\varphi(x), u_y|_{y=0}=\psi(x), x\in\Bbb R$.
If I use method of characteristics, $\xi=x+2y-\ln\frac{1+y}{1-y}$, $\eta=y$, then $2(\eta^2-1)^2(1+\eta^2)u_{\xi\eta}+4\eta u_\xi+2\eta(\eta^2-1)u_\eta=0$ And then let $z=\xi+\eta, w=\xi-\eta$, we have $u_{zz}-u_{ww}+f(z,w)u_z+g(z,w)u_w=0$. Oh. I still could not solve it...Since $f,g$ are not constant.
The coefficients depend only on $y$! How to use it?
