= Curvilinear reduction to the one-dimensional wave equation
{title2=$r=\sqrt x,\quad s=y^{3/2}/3$}
In $x,y>0$, the change of variables in the title reduces
$$
-xyu_{xx}+u_{yy}=\frac12\left(yu_x+\frac{u_y}{y}\right)
$$
to the <one-dimensional wave equation> $U_{ss}-U_{rr}=0$. Indeed $u_{xx}=U_{rr}/(4x)-U_r/(4x^{3/2})$ and $u_{yy}=yU_{ss}/4+U_s/(4\sqrt y)$; the first-derivative terms cancel exactly against the right-hand side. The independent <characteristic coordinates> $\xi=r+s$, $\eta=r-s$ then give $V_{\xi\eta}=0$ and $u=g(\xi)+h(\eta)$. Two characteristic families alone generally reduce only the principal part; the cancellation of lower-order terms is the extra feature that gives this simple exact solution.
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