 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-2/7a/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The only equilibrium is

(0, 0)

(\overset{x}{˙} \overset{y}{˙}) = (0 - 1 10) (x y),

\pm i

, so the linearized system has a centre.

The exact radial identity from part (i) gives

\overset{r}{˙} = k r^{3} .

Thus the origin is asymptotically stable for

k < 0

, neutrally stable for

k = 0

, and unstable for

k > 0

. Moreover,

\dot{θ} = \frac{x y ˙ - y x ˙}{r ^{2}} = - 1,

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