 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-2/3b/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

\frac{d x}{d s} = - y, \frac{d y}{d s} = x, \frac{d u}{d s} = 0.

Along a characteristic,

\frac{d}{d s} (x^{2} + y^{2}) = 2 x (- y) + 2 y x = 0.

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)

. Hence

u (x, y) = f (x^{2} + y^{2}) .

The characteristic sketch is the family of concentric circles centred at the origin.

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