I came across the following coupled system of 2nd order ODEs from the appendix of this paper :
$$r^2 f_{rr} + 2r f_r - \frac{r}{2n+1}h_r - (n+1)(n+2)f = 0$$ $$r^2 g_{rr} + 2r g_r + \frac{r}{2n+1}h_r - n(n-1)g + \frac{n}{2n+1}h = 0$$ $$(n+1)r f_r - nrg_r + (n+1)(n+2)f + n(n-1)g = 0$$
Here, $f, g, h$ are functions of r, taking finite values as $r \rightarrow \infty$ and going to zero as $r \rightarrow 0$. $p_r, p_{rr}$ denotes the first and second derivative respectively.
They have solved for this system analytically.
I understand I would have to formulate a polynomial ansatz to arrive at the solution for this equation, but it is unclear how one should make an educated guess about it. Is there a way to find out?