# Series solution of an ODE with nonpolynomial coefficients

Basically, I have a second-order differential equation for $$g(y)$$ and I want to obtain a series solution at $$y=\infty$$ where $$g(y)$$ should vanish. That would be easy if the ODE contains polynomial coefficients, hence the Frobenius method can be used. But in my case, the coefficients are not polynomial because of the presence of powers proportional to $$p$$ (can take positive non-integer values) and because of the form of the function $$r(y)$$. The ODE reads

$$r(y)^2 g''(y)+(2\,r(y) - p\,r(y)^{1 - p})r'(y) g'(y) - l (l + 1) g(y)=0$$ where the prime denotes differentiation with respect to $$y$$, and $$r(y)$$ is given as $$r(y)=\sqrt{-5 + y^2 + \frac{3\cdot 2^{1/3}}{(2 + 10 y^2 - y^4 + \sqrt{64 y^2 + 48 y^4 + 12 y^6 + y^8})^{1/3}} - \frac{6\cdot 2^{1/3}y^2}{(2 + 10 y^2 - y^4 + \sqrt{64 y^2 + 48 y^4 + 12 y^6 + y^8})^{1/3}} + \frac{3 (2 + 10 y^2 - y^4 + \sqrt{64 y^2 + 48 y^4 + 12 y^6 + y^8})^{1/3}}{2^{1/3}}}$$

My question would be, am I allowed to get the respective Taylor series expansion of the coefficients in front of $$g''(y)$$ and $$g'y)$$ in the ODE to simplify the problem, and then perform a change of variable to express the coefficients in their polynomial form? Any hint in solving this problem is very much appreciated. Thanks