Examples for Hilbert's Quote
Hilbert once said, “The art of doing mathematics consists in finding that special case which contains all the germs of generality.”
What would be (relatively) simple examples?
Solution 1:
$$x_{n-1} =\frac{\alpha+\beta x_n +\gamma x_{n-1} +\delta x_{n-2}}{A+Bx_{n}+Cx_{n-1}+Dx_{n-2}}, n= 0, 1, ...,$$
where the parameters $\alpha, \beta, \gamma, \delta, A, B,C, D$ are non-negative real numbers and the initial conditions $m$ are arbitrary non-negative real numbers such that the denominator is always positive.
We are primarily concerned with the boundedness nature of solutions, the stability of the equilibrium points, the periodic character of the equation, and with convergence to periodic solutions including periodic trichotomies.
If we allow one or more of the parameters in the equation to be $0$, then we can see that the equation contains
$$(2^4 -1)(2^4 -1)= 225 $$
special cases, each with positive parameters and positive or non-negative initial conditions.
According to David Hilbert “The art of doing mathematics consists in finding that special case which contains all the germs of generality" and according to Paul Halmos 'The source of all good mathematics is the special case, the concrete example"
The special case of this equation contains a lot of the germs of generality of the theory of difference equations of order greater than one about which, at the beginning of the third millennium, we know no surprisingly little. The mathematics behind the special cases of this equation is also beautiful, surprising, and interesting.
The methods and techniques we develop to understand the dynamics of various special cases of rational difference equations and the theory that we obtain will also be useful in analysing the equation in any mathematical model that involves difference equation.
Solution 2:
Integers. Many theorems of rings, for example, give meaningful non-trivial results. Think of the abstract theory of prime ideals. Take matrices as a generalization - and even that can be considered a nice "special case" in itself. Then think operator algebras... I mean so many structures in algebra just generalize what you may do with individual numbers.