Characteristic normal form ($U_{\phi \psi }+\text{lower-order terms}=F$)

For a hyperbolic partial differential equation in two variables, two independent characteristic coordinates remove both pure second partial derivatives, leaving a nonzero mixed second partial derivative. A nonzero Jacobian determinant is essential: two characteristic invariants that become dependent do not give a valid coordinate transformation.

In $x,y>0$, the change of variables in the title reducesto the one-dimensional wave equation $U_{ss}-U_{rr}=0$. Indeed $u_{xx}=U_{rr}/(4x)-U_r/(4x^{3/2})$ and $u_{yy}=y{U}_{ss}/4+U_s/(4\sqrt{y})$; 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.