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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 14C / b / i / Solution

Substitution gives

(g \tilde{n})^{'} = - (μ + r) \tilde{n}

, hence

\tilde{n} (s) = \frac{C}{g ( s )} exp [- \int_{0}^{s} \frac{μ ( u ) + r}{g ( u )} d u] .

The boundary condition yields the Euler-Lotka equation

1 = \int_{0}^{\infty} \frac{b ( s )}{g ( s )} exp [- \int_{0}^{s} \frac{μ ( u ) + r}{g ( u )} d u] d s .

Solved by gpt-5.6-sol high.

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