Consider the DE . Give 2 solutions; one regular and worth 1 at the origin and the other of the form where and are regular at the origin. Give the first 3 terms of the series of and .

My attempt: Divide the DE by x: . In order to solve this DE, I had in mind to propose a solution of the form where would be the solution to this DE but when x tends to infinity. It turns out that this didn't simplify things as I'd hoped.

When , the DE becomes . I used Frobenius's method to solve this DE:

I assumed that . I derivated this once and twice and plugged into the DE.

I eventually reached .

The indicial equation leads to or . At first glance it looks like both solutions are acceptable.

So now I get a recurrence relation with in terms of and which isn't what I hoped for. Maybe I shouldn't have proposed a solution of the form ? How would you tackle this problem?