Solution
= Solution
This <birth-death process> has birth rates $b_n=\alpha+\beta n$ and death rates $d_n=\gamma n(n-1)$. With $P_{-1}=0$, its <birth-death master equation> is
$$
\boxed{
\frac{dP_n}{dt}
=[\alpha+\beta(n-1)]P_{n-1}
+\gamma(n+1)nP_{n+1}
-[\alpha+\beta n+\gamma n(n-1)]P_n.}
$$
The first two terms are inflow by a birth from $n-1$ and a death from $n+1$; the bracketed term is the total outflow from $n$.