Solution
= Solution
The <principal symbol> of
$$
u_t-(x^2-1)u_{xx}=u_t+(1-x^2)u_{xx}=0
$$
is $p(x,t;\xi,\tau)=(1-x^2)\xi^2$. For a regular curve $s\mapsto(x(s),t(s))$, the <characteristic curve> equation is
$$
(1-x^2)\dot t^{,2}=0.
$$
Away from $x=\pm1$ this gives $t=\text{constant}$. The two degenerate lines $x=1$ and $x=-1$ are also characteristic. These are the characteristic curves, apart from reparametrization and pieces joined at the degenerate lines.