Singular points and finding linearly independent solutions

The given equation is

1. How would I go proving a singular point that is regular?

I have the textbook in front of me and it doesn't help me at all. I was able to find x=0, but the question asks me to show it. So I did by making y'' equaling to 1 and divided the variable to the rest. I'm just confused if that's good enough. In my end, I don't think it is because they say show that it's a regular singluar point. I'm just a bit confused if I should show more or if what I just mentioned is ok.

2. Find two linearly independent solutions of the form . Using the recurrence relation.

Ok so I changed the given y to

Then I found the first and second derivatives.

When I plug it in back into the original equation, I get:

+ + -

Am I doing this correct so far? I would like to make sure before I continue and would it be easier to take out , , or does it not matter?

Thanks. I'll come back after seeing if I'm doing the first part correct and post my solution to see if y'all agree.