I want to get the stationary state of the simple birth-death process using the Fokker-Planck expansion. This describes a population growing from births at rate $\lambda$ and shrinking from deaths at rate $\sigma$. The governing equations for the probabilities $P(n)$ that the population has size $n = 0,1,2,\dots$ are $$0 = \lambda P(n-1) + \sigma[n+1]P(n+1) - [\lambda + \sigma n]P(n).$$ Of course, this can be solved exactly for the Poisson distribution $$ P(n) = \frac{(\lambda/\sigma)^n}{n!}e^{-\lambda/\sigma},$$ but I would like to solve it approximately to gain familiarity with the Fokker-Planck equation.
To do this, I have attempted to treat $n$ as a continuous variable and expanded to second order in it, obtaining
$$ 0 \approx \lambda\Big[P - P' + \frac{1}{2}P''\Big] + \sigma\Big[ nP + (nP)' + \frac{1}{2}(nP)''\Big]- \Big[\lambda + \sigma n\Big]P$$ which simplifies to $$ 0 = \sigma P(n) + (\sigma [ n+1] -\lambda )P'(n) + \frac{1}{2}(\lambda + \sigma n)P''(n).$$
Unfortunately I do not know how to solve this non-linear ODE. Probably since $n\gg 1$ in the Fokker-Planck approximation, I can drop the $1$ in the second term and divide by $\sigma$ to obtain $$ 0 = P(n) + \Big(n - \frac{\lambda}{\sigma} \Big)P'(n) + \frac{1}{2}\Big(n + \frac{\lambda}{\sigma}\Big)P''(n).$$
Provided I'm on the right track (am I?), the solution of this should approximate the Poisson distribution. Can anyone teach me how to show this?