(1) Since or are not analytic at , is a singular point of the differential equation. Using Frobenius' Theorem, we must check that and are both analytic at . Since and are analytic at , is a regular singular point for the differential equation From the result of Frobenius' Theorem, we may assume that has a solution of the form which converges for where and are constants that will be determined later.
(2) Substituting into , we get that
(3) In this step, we will use the equation above to find the indicial roots and the recurrence relation of the differential equation.
(a) From the equation above, we know that the indicial roots of the differential equation are and .
(b) From the constant term in the series above, we know that
(c) From the series above, we find that the recurrence relation is
(4) The general solution to is
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