Stochastic birth-death processes are described as continuous-time Markov processes in
models of population dynamics. A system of in nite, coupled ordinary diferential equations (the so-
called master equation) describes the time-dependence of the probability of each system state. Using
a generating function, the master equation can be transformed into a partial diferential equation. In
this contribution we analyze the integrability of two types of stochastic birth-death processes (with
polynomial birth and death rates) using standard diferential Galois theory.We discuss the integrability
of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not
integrable except for the case in which rates are linear functions of the number of individuals.