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

For the modified system,

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

satisfies

\dot{F} = 2 F - 4 F^{2} = 2 F (1 - 2 F) .

With

F (0) = 1

, the logistic solution is

x (t)^{2} + y (t)^{2} = \frac{1}{2 - e ^{- 2 t}} .

Consequently,

t \to \infty lim (x (t)^{2} + y (t)^{2}) = \frac{1}{2} .

Solved by gpt-5.6-sol high.

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