I have a 2nd order homogenous ODE: $$py'' + (2l+2-2p)y' + 2(n-l-1)y = 0$$ where y is a function of p and n and l are variables
Its solution is the associated Laguerre polynomials $L_{n-l-1}^{2l+1}(2p)$
I have expanded out the solution to try and 'meet in the middle' to get this:
$$2py'' + (2l+2-2p)y' + (n-l-1)y = 0$$
But I just can't work out how to rearrange the first one to achieve the second.
If anyone would like to know where this equation comes from, it is the 'main' part of the radial equation for the hydrogen atom (i.e. it is the radial equation with the asymptotic solutions removed). In most textbooks it is denoted $v(\rho)$.
Edit: Thanks to Robert Israel's helpful answer I'm aware that the second differential equation above does not have the solution of the Laguerre polynomials $L_{n-l-1}^{2l+1}(2p)$. If anyone could tell me the correct differential equation that does have those solutions I would be very grateful.
The two equations are not the same, but they are related. Using Maple I get $L^{2l+1}_{n-2}(2p)$ as a solution of the first, but $L^{n/2-1/2-l/2}_{l}(p)$ as a solution of the second. [Caution: conventions may difer]