Laguerre's differential equation

Laguerre's differential equation is a second order linear Ordinary differential equation of a function $x(t)$ :

t \frac{d^{2}x}{dt^{2}}+(1-t) \frac{dx}{dt}+nx=0

It has nonsingular solution only for $n\in \mathbb{N}$ , and those solutions are the Laguerre polynomials. This equation can be generalized to

t \frac{d^{2}x}{dt^{2}}+(\alpha+1-t) \frac{dx}{dt}+nx=0

for some $\alpha \in \mathbb{R}$ . Nonsingular solutions to this equation are known as the associated Laguerre polynomials.