I have the following ODE (which comes from a physics book):
$$\frac{d \chi^2}{d \eta^2}+(k^2+C(\eta) m^2)\chi(\eta) = 0,$$
where $C(\eta)= A+B\tanh(\rho\eta)$ and $k,m,A,B,\rho$ are constants.
The book hints that it is solvable in terms of hypergeometric functions and presents a solution for it.
I would like to check the solution presented in the book but I'm having trouble finding the appropriate transformations that put this ODE in the form of a hypergeometric equation. How can I find it?
Hint:
Follow the method in http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=276:
Let $\begin{cases}t=\dfrac{1-\tanh(\rho\eta)}{1+\tanh(\rho\eta)}\\x=\chi t^{\pm\frac{\sqrt{m^2(B-A)-k^2}}{2\rho}}\end{cases}$