Problem with solving the Colatitude equation

[mavyn]

1. The problem statement, all variables and given/known data
Show that the priciple quantum number n limits the values of the angular momentum quantum number l such as

l <= n-1

Calculate the degeneracy of energy levels of the hydrogen atom when only the Colomb force is taken into account.

The attempt at a solution
I wrote the 3D-Schrödinger equation (in spherical coordinates) and solved it by seperating the variables in the form

Psi(r,phi,theta) = R(r) * P(theta) * F(phi) -> so i have 3 equations for 3 quantum numbers.

Then I solved the azimuthal equation to get to constant Cphi, which is

Ctheta = -ml^2 (ml.. magnetic quantum number)


So I have now the Colatitude equation:

sin(theta)/P * d/dtheta [ sin(theta) dP/dtheta] + Cr sin^2(theta) = -Cphi with Cphi = -ml^2

So far I'm right, I think..

I know that, if I solve this equation it will show that "l <= n-1". I also know that I have to do this by using polynomial expansion.
But I don't know how to do this here.

I hope somebody can help me! Thx