After searching for a while I finally found a detailed solution to this general Legendre equation.
So, I guess this question is answered!
The source for the solution is this:
http://www.physicspages.com/2011/03/22/associated-legendre-functions/
EDIT: As of February 12, 2018 the above link doesn't work. Use this instead: https://web.archive.org/web/20170509204654/http://www.physicspages.com:80/2011/03/22/associated-legendre-functions/