The given equation is: $$x^2y''+xy'+\left(x-5\right)y=0$$
Using the method of frobenius where: $$y = \sum_{n=0}^\infty A_nx^{n+r}$$
$$y' = \sum_{n=0}^\infty A_n(n+r)x^{n+r-1}$$
$$y'' = \sum_{n=0}^\infty A_n(n+r)(n+r-1)x^{n+r-2}$$ I substituted the y, y', y'' into the original equation:
$$\sum_{n=0}^\infty A_n(n+r)(n+r-1)x^{n+r} + \sum_{n=0}^\infty A_n(n+r)x^{n+r} + \sum_{n=0}^\infty A_nx^{n+r+1} - 5\sum_{n=0}^\infty A_nx^{n+r}=0$$
After I re-indexed the the third summation, I simplified it to:
$$A_2 x^{r+1} +\sum_{n=0}^\infty (A_n(n+r)^2-5A_n+A_{n+1})x^{n+r} = 0 $$
Is this correct so far? If so, what do I need to do to find the series solution?
Thank you