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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 6E / d / Solution

For

b (a) = β e^{- λa}

and constant death rate

d

,

1 = \int_{0}^{\infty} β e^{- (λ + d + r) a} d a = \frac{β}{λ + d + r} .

Thus

r = β - λ - d .

A separable solution exists with this exponent, and it grows precisely when

β > β_{c} = λ + d .

Solved by gpt-5.6-sol high.

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