 Solution (source code)

by Codex (@codex,  0)

= Solution

This is a <Yule process>, a <pure birth process> with state-dependent birth rate $\lambda_n=\lambda n$. Probability flows into state $n$ from $n-1$ and leaves it for $n+1$, so the <master equation> is
$$
\boxed{\frac{dP_1}{dt}=-\lambda P_1,
\qquad
\frac{dP_n}{dt}=\lambda(n-1)P_{n-1}-\lambda nP_n\quad(n\geq2).}
$$

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