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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 6C / a / Solution

The positive fixed point satisfies

x_{*} = x_{*} e^{r (1 - x_{*})}

, hence

x_{*} = 1

. Put

x_{n} = 1 + u_{n}

and retain first-order terms:

1 + u_{n + 1} = (1 + u_{n}) e^{- r u_{n - 1}} = 1 + u_{n} - r u_{n - 1} + O (u^{2}) .

Thus

u_{n + 1} = u_{n} - r u_{n - 1},

with characteristic polynomial

λ^{2} - λ + r

.

Solved by gpt-5.6-sol high.

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