Solution
= Solution
Starting with one individual, let $S$ be the first event time, of density
$(\lambda+\mu)e^{-(\lambda+\mu)s}$. At that event there is a death with conditional probability $\mu/(\lambda+\mu)$, causing extinction immediately, or a birth with probability $\lambda/(\lambda+\mu)$, leaving two independent descendant populations. The branching property therefore gives
$$
\boxed{h(t)=\int_0^t e^{-(\lambda+\mu)s}
\{\mu+\lambda h(t-s)^2\}\,ds}.
$$