Second Order ODE with Exponential Coefficients

[thatboi]

Hi all,
I have another second order ODE that I need help with simplifying/solving:
$p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0$
where $\gamma,A,F$ can all be assumed to be nonzero real numbers and $D$ is a purely nonzero imaginary number.
Any help would be appreciated!

 

[pasmith]

Set $t = e^{\gamma x}$. Then $$\gamma^2(1-t)t^2 \frac{d^2p}{dt^2} + \left(\gamma^2(1-t)t - \frac{\gamma D}{A} t^2\right) \frac{dp}{dt} - F(1-t)p = 0.$$ This looks like it should be solvable by Frobenius' method.