= Quadratic characteristic coordinates for a variable-speed wave equation
{title2=$u_{xx}-x^2u_{yy}-u_x/x=0\iff U_{\xi\eta}=0,\quad\xi=x^2-2y,\ \eta=x^2+2y$}
On $x>0$, the <linear partial differential equation> $u_{xx}-x^2u_{yy}-u_x/x=0$ has <characteristic curves> $y\pm x^2/2=\text{constant}$. The <change of variables> $\xi=x^2-2y$, $\eta=x^2+2y$ has <Jacobian determinant> $8x\ne0$. The <chain rule> gives
$$
u_{xx}-x^2u_{yy}-u_x/x=16x^2U_{\xi\eta}.
$$
Thus its <canonical form of a second-order partial differential equation> is $U_{\xi\eta}=0$, and its general twice differentiable solution is $u=f(x^2-2y)+g(x^2+2y)$. The first derivative term exactly cancels the derivatives of the coordinate coefficients.
More generally, for positive differentiable $a(x)$, choose $s(x)$ with $s'(x)=a(x)$. Then the same cancellation gives
$$
u_{xx}-a(x)^2u_{yy}-\frac{a'(x)}{a(x)}u_x
=a(x)^2(U_{ss}-U_{yy}).
$$
The <characteristic> coordinates are $s\pm y$. The quadratic example is $a(x)=x$, $s=x^2/2$, with both coordinates multiplied by two.
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