= Logarithmic characteristic coordinates
{title2=$\xi=y+\ln|x|,\quad\eta=y-\ln|x|$}
For the Euler-type operator $-x^2\partial_x^2-x\partial_x+\partial_y^2$, set $z=\ln|x|$ on either half-plane $x\ne0$. 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 $-x^2u_{xx}-xu_x+u_{yy}=c$ 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 $x=0$.
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