Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 105 / 1 / b / i / Solution

The principal symbol ofis $p(x,t;\xi ,\tau )=(1-x^2){\xi }^2$. For a regular curve $s\mapsto (x(s),t(s))$, the characteristic curve equation isAway from $x=\pm 1$ 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.