 Solution

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

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

Take

F = x^{2} + y^{2} .

Then

\dot{F} = 2 x \overset{x}{˙} + 2 y \overset{y}{˙} = 2 k (x^{2} + y^{2})^{2} = 2 k F^{2} .

With

F (0) = 1

, separation gives

- \frac{1}{F} = 2 k t - 1,

and hence

F (t) = \frac{1}{1 - 2 k t}

on its maximal interval of existence.

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