I was studying Non-Equilibrium phase transition models and got stuck on the very first simple model that I was looking at. The Population of species is given by $n(t)$ the transition rates are
$P(n:t;n+1, t+\Delta t) = \lambda n$ $P(n:t,n-1, t+\Delta t) = n$.
The resulting master equation and dynamics is as follows, We use the notation $P_n(t)$ for the probability of population $n$ at time $t$.
$\frac{dP_n(t)}{dt} = \lambda (n-1) P_{n-1}(t-\Delta t) + (n+1) P_{n+1}(t-\Delta t) - (1 + \lambda) P_n(t - \Delta t)$
The moment generating function $g(x,t) = \sum_n x^n P_n(t)$. We obtain the differential equation below,
$\frac{\partial g}{\partial t} = (1 - x)(1-\lambda x) \frac{\partial g}{\partial x}$
I get all this, the solution of the above using separation of variables I get is very different from the one presented in the book.
$g(x,t) = \frac{(1-\lambda x)e^{(1-\lambda t)} - (1-x)}{(1-\lambda x)e^{(1-\lambda t)} - \lambda(1-x)}$
Clearly, the above solution is not obtained by separation of variables.
Help is appreciated and thanks in advance.