The given equation is

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

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:

+ + -

Ok, I understand up to this part. Now I'm just confused after this, but this is what I got. I took out

+ + -

Then I tried to get and changed some of the n values to and similar to what my professor did.

+

Then I tried to find the r by:

I got r = 1/2 and r = -1.

Is this correct so far? When my professor went over a homework question similar to this problem, he said that the r's would be -1 and -1/2. I just don't see where the -1/2 came from and even so. When I plug -1 , I get zero in the third term which ruins it unless that's ok?

I know what to do pass this part, but it's very confusing. I spent more than 5 hours trying to do this and I know I'm doing something wrong or thinking of something else. Please help me lead to the right direction.