= Characteristic coordinates for a linearly changing PDE type
{c}
{title2=$yu_{xx}-u_{yy}=0,\quad\phi=3x-2y^{3/2},\ \psi=3x+2y^{3/2}$}
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> gives
$$
yu_{xx}-u_{yy}=36y\,u_{\phi\psi}+\frac3{2\sqrt y}(u_\phi-u_\psi),\qquad u_{\phi\psi}=\frac{u_\psi-u_\phi}{6(\psi-\phi)}.
$$
The 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.
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