I'm trying to derive a Fokker Planck equation for a stochastic process. Consider a population of size n, in which births and deaths are random events which occur at rates $\lambda$ and $\mu$ respectively. Let $P(n,t)$ denote the probability of the population size being $n$ at time $t$. Then, for a time step $\Delta t$ which is "sufficiently" small that an individual experiences only one event, the transition rates between states are $$W^+(n-1) = \lambda \Delta t (n-1)$$
$$W^-(n+1) = \mu \Delta t(n+1)$$
$$W^+(n) = \lambda \Delta t n $$ $$W^-(n) = \mu \Delta t n$$
These lead to the master equation $$\frac{\partial P(n,t)}{\partial n} = W^+(n-1) P(n-1,t) + W^-(n+1) P(n+1,t) - W^+(n) P(n,t) - W^-(n) P(n,t).$$ Using the first and second moments, $$\langle n \rangle = (\lambda - \mu)\langle n \rangle, $$ $$\langle n^2 \rangle = (\lambda + \mu)\langle n \rangle + 2(\lambda - \mu)\langle n^2 \rangle$$ I found the Fokker Planck equation for this process to be $$\frac{\partial P(n,t)}{\partial t} = - \frac{\partial}{\partial n} (\lambda-\mu) n P(n,t) + \frac{1}{2} \frac{\partial^2}{\partial n^2}(\lambda + \mu)n P(n,t).$$
When I solve this Fokker Planck equation I arrive at $$P(n) = \frac{1}{n} exp\left(\frac{\lambda - \mu}{\lambda + \mu}n + C \right)$$ where $C$ is a constant from integration. Using this formulation, the probability of extinction is undefined which is obviously incorrect. Is my error in my formulation of the transition rates, or in the derivation of the master equation?