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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 7A / i / Solution

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.

Solved by gpt-5.6-sol high.

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