 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/6e/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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 .

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