Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 3B / Solution

The method of characteristics givesAlong a characteristic,Thus the characteristics are circles $x^2+y^2=C$, traversed counterclockwise, and $u$ is constant on each circle. Their intersections with the $x$-axis have $x^2=C$, where the boundary value is $f(C)$. HenceThe characteristic sketch is the family of concentric circles centred at the origin.