Characteristic coordinates for a linearly changing PDE type ($y{u}_{xx}-u_{yy}=0,\varphi =3x-2y^{3/2},\psi =3x+2y^{3/2}$)

This partial differential equation has discriminant $y$, so it is a hyperbolic partial differential equation above $y=0$ and an elliptic partial differential equation below it. In its hyperbolic region the displayed characteristic coordinates have Jacobian determinant $18\sqrt{y}$. The chain rule givesThe first derivative terms arise from differentiating the coordinate coefficients and cannot be discarded. The coordinate change degenerates at the line where the two characteristic directions merge.