Given a differential equation:
$$f''(x) + (1-B^2x^2)f(x) - f(x)^3 = 0$$
subject to the boundary conditions
$$f(x=0) =0 \qquad f(x\to\infty) = \gamma_A$$ where $\gamma_A$ is some constant between 0 and 1.
As it stands, this is a non-linear, 2nd order ODE. My undergraduate experience had shockingly little exposure to these type of equations, even if I was to omit the non-linear term.
Dropping the final $f(x)^3$ term I believe leads to some kind of Whittaker(?) type equation with some hypergeometric functions, but I have not encountered these before. I am not sure if there even exists an obvious way to solve this analytically, and I am not even certain if I can solve this numerically with the boundary condition tending to infinity - the non-constant $x^2f(x)^2$ makes incorporating this difficult. How would one begin to approach a problem like this? I would be very interested to hear the approach that one might take to even to begin solving.
My instinct suggests that this should appear to be a tanh-like solution, beginning at 0 and having an asymptote towards $\gamma_A$ as $x$ is increased. In the full, non-linear equation, taking the case $B=0$ and $\gamma_A =1 $ in fact leads to a tanh solution, but it is not clear what the analytic solution is when $\gamma_A \neq 1 $, although this is easier to solve numerically for. Any advice would be appreciated!