Solve DE for theta component (hydrogen WF)

[Andrew Deleonardis]

The particular equation I would like to see solve is:
$\sin\theta\frac{d}{d\theta}(\sin\theta\frac{d\Theta}{d\theta})+\Theta(l(l+1)\sin\theta-m^2)$

The solution for this equation is the following associated laguerre polynomial:
$P^m_l(\cos\theta)=(-1)^m(\sin\theta)^m\frac{d^m}{d\cos\theta^m}\bigg(\frac{1}{2^ll!}\frac{d^l}{d\cos\theta^l}(cos^2\theta-1)^l\bigg)$

This equation is involved in solving the Schrödinger equation for the hydrogen atom.
Even though I already know the answer, I would like to know HOW to solve it

 

[blue_leaf77]

Although never tried it myself, associated Legendre differential equation can be solved by power series expansion. For the ordinary Legendre differential equation see this. I guess you should be able to get the idea from that link when applied to the associated Legendre equation.
By the way, no one can solve the equation you have there until you add equal sign and some terms in the other side.

 

[Andrew Deleonardis]

Although never tried it myself, associated Legendre differential equation can be solved by power series expansion. For the ordinary Legendre differential equation see this. I guess you should be able to get the idea from that link when applied to the associated Legendre equation.
By the way, no one can solve the equation you have there until you add equal sign and some terms in the other side.

Oh oops, it's meant to equal zero, I forgot

 

[Andrew Deleonardis]

The particular equation I would like to see solve is:
$\sin\theta\frac{d}{d\theta}(\sin\theta\frac{d\Theta}{d\theta})+\Theta(l(l+1)\sin\theta-m^2)$
Edit: =0

The solution for this equation is the following associated laguerre polynomial:
$P^m_l(\cos\theta)=(-1)^m(\sin\theta)^m\frac{d^m}{d\cos\theta^m}\bigg(\frac{1}{2^ll!}\frac{d^l}{d\cos\theta^l}(cos^2\theta-1)^l\bigg)$

This equation is involved in solving the Schrödinger equation for the hydrogen atom.
Even though I already know the answer, I would like to know HOW to solve it