Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 28K / b / iii

Separating variables in
$h^{\prime }=(h-1)(\lambda h-\mu )$ with $h(0)=0$ gives the extinction probability of a linear birth-death processFor $\lambda >\mu$ this tends to $\mu /\lambda$; for $\lambda <\mu$ it tends to one, as expected.