Solve the PDE $x^2U_{xx}+U_{yy}+U\log U=0$

Question

Solve the PDE $x^2U_{xx}+U_{yy}+U\log U=0$

my attempt:

this is elliptic and $\lambda=\pm \frac{i}{x}$

so $\frac{dy}{dx}=\frac{i}{x}, \frac{dy}{dx}=\frac{-i}{x}$

from here to how to processed