I'm trying to solve the following PDE
$$y^2 u_{xx} - u_{yy} = 0.$$
I've found its canonical form as $$u_{\epsilon \eta} = \frac{ u_\epsilon - u_\eta}{4 (\epsilon - \eta) } .$$
However, I'm having trouble in solving this PDE. I've tried solving with mathematica, and that gave me
$$u = \frac{ x + y^3}{ 3} ,$$ but, I need to know how to obtain such a result from the canonical form of the given PDE.
Edit:
The coordinate transformations are $$\epsilon = x + y^2 / 2 \qquad \eta = x - y^2 /2$$