Solving for the recursion relation for the expansion coefficients of the asymptotic expansion of an ODE

Question

I want to solve for the asymptotic solution of the following differential equation

$$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$

as $y\rightarrow \infty$, where $p>0$. I did the standard way by obtaining a series solution by the Frobenius method prescription in the form

$$R(y)=\sum_{n=0}^\infty \frac{a_{n}}{y^{n+k}}$$ where $k=l+1$ is the indicial exponent. I had difficulty finding, by hand, for a recurrence relation for the coefficients $a_n$ for arbitrary value of the parameter $p$. Right now, I am just doing the brute force method of solving individual $a_n$ for every value of $p$.

But I am just wondering whether the recurrence relation is possible to obtain using Mathematica routine. Any help is appreciated.

Answer 1

For finding asymptotic solutions you can use AsymptoticDSolveValue function.

eqn = (y^2 + 1) R''[y] + y (2 - p (b0 Sqrt[y^2 + 1])^(-p)) R'[y] - l (l + 1) R[y]
AsymptoticDSolveValue[eqn == 0, R[y], {y, \[Infinity], 1}]

Unfortunately this does not work for arbitrary p, thus you will not get a general answer. However, you can just try it for different values of p, sometimes it gives result, sometimes not, e.g.,

AsymptoticDSolveValue[eqn == 0 /. {p -> 5/2}, R[y], {y, \[Infinity], 1}]

I hope that helps.