= Solution
The <diffusion approximation of a birth-death process> uses the first two jump moments. For these unit upward and downward jumps they give
$$
\boxed{
u(x)=\alpha+\beta x-\gamma x(x-1),\qquad
D(x)=\frac12\{\alpha+\beta x+\gamma x(x-1)\}.}
$$
The factor $1/2$ in $D$ is required because the equation is written with $+\partial_x^2(DP)$, rather than $\tfrac12\partial_x^2(BP)$.
Assuming the boundary terms vanish, integration by parts gives
$$
\begin{aligned}
\frac d{dt}\langle x\rangle
&=\int x\left[-\frac{\partial(uP)}{\partial x}
+\frac{\partial^2(DP)}{\partial x^2}\right]dx\\
&=\int u(x)P(x,t)\,dx=\langle u(x)\rangle.
\end{aligned}
$$
Consequently
$$
\boxed{
\frac d{dt}\langle x\rangle
=\alpha+\beta\langle x\rangle
-\gamma\langle x(x-1)\rangle,}
$$
which is precisely the discrete first-moment equation in part (b).
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