I am solving a (infinitely long) differential equation which has the solution

$$

y(r)=-frac{c}{5}+frac{l^4c^3}{20r^5}+frac{l^{6}c^5}{16r^9}+mathcal{O}(l^8),

$$

where I am not sure about the sign of the last term. Additionally, I know that this solution is a series of the form

$$

y(r)=-left(frac{c}{r}right)cdotp_2F_1(d,e;f;z),

$$

Where $_2F_1$ is a hypergeometric series. My question is: how can I make mathematica find the exact form of the hypergeometric series, based on the first few terms?

Thanks in advance!