I remember one day having stumbled upon a nice online resource where one could look for solutions to very general ODEs (or at least the literature names thereof), but unfortunately I forgot its location... So, where can one go looking to e.g. figure out that
$$\frac{d}{dr}\left[(1-r^2)\frac{d}{dr}f(r)\right] + l(l+1)f(r) = 0$$
is the Legendre differential equation, especially if one doesn't know about this particular representation?
You can also try http://eqworld.ipmnet.ru For example for your request: http://eqworld.ipmnet.ru/en/solutions/ode/ode0223.pdf