= Solution
Differentiate $\langle f(n)\rangle=\sum_{n\geq0}f_np_n$ and substitute the three master equations. Reindexing every gain term by its source state gives one contribution per possible transition:
$$
\begin{aligned}
\frac d{dt}\langle f(n)\rangle
={}&b\sum_{n=0}^\infty(f_{n+1}-f_n)p_n\\
&-d\sum_{n=2}^\infty(f_n-f_{n-2})np_n
-D(f_1-f_0)p_1.
\end{aligned}
$$
This is the <Markov jump-process generator>[generator identity for a continuous-time Markov chain] applied to the test function $f$.
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