I came across this equation in my research (related to reaction diffusion system):
$$\frac{d^2y}{dr^2}+B\operatorname{sech}^2(r) \frac{dy}{dr} + Cy = 0$$
where $B$ and $C$ are constants. Can it be solved analytically?



Non-constant-coefficient second-order linear ODE
Making the change of the independent variable $x=e^{it}$, we obtain
$$x^2w''+xw'+\frac{4iBx^3}{(x^2+1)^2}w'-Cw=0.$$
This equation has $2$ regular singularities $0,\infty$, and irregular
at $\pm i$. Therefore
for generic $B$ and $C$ its solutions cannot be expressed in terms of functions
of hypergeometric type, or any other usual special functions.


