Logarithmic characteristic coordinates ($\xi =y+\ln |x|,\eta =y-\ln |x|$)

For the Euler-type operator $-x^2{\partial }_x^2-x{\partial }_x+{\partial }_y^2$, set $z=\ln |x|$ on either half-plane $x\ne 0$. The chain rule gives $x{\partial }_x={\partial }_z$ and $x^2{\partial }_x^2+x{\partial }_x={\partial }_z^2$. The operator becomes ${\partial }_y^2-{\partial }_z^2$. Its characteristic coordinates $\xi =y+z$, $\eta =y-z$ reduce it further to $4{\partial }_{\xi }{\partial }_{\eta }$. Thus the constant-forcing equation $-x^2{u}_{xx}-x{u}_x+u_{yy}=c$ has solutions $u=(c/4)\xi \eta +g(\xi )+h(\eta )$, with arbitrary sufficiently differentiable functions $g,h$. The logarithmic change of variables cannot by itself describe smooth passage across $x=0$.