Here's the link explaining the method of Frobenius with a double root I am referring to: http://www.creighton.edu/fileadmin/user/CCAS/departments/Math/docs/DE/series_solutions/Equal_Roots_Example.pdf
Why do they take $\frac{\text{d}a_n}{\text{d}r}$? How does this guarantee the solution they find will actually work?