Special functions and diff eq's ...... They're are all these methods of dealing with linear second order diff eq's:
generating function;
recurrence relation;
Rodrigues differential form;
Schlafi integral form;
associated form;
second form;
shifted form;
series form;
you can use on differential equations with special names:
Airy; 
Bessel; 
Chebyshev; 
Gauss hypergeometric;
Hermite;
Jacobi;
Laguerre;
Legendre
There are at least 100 ideas in these two little lists, and apparently one can start from anything in the first list and derive any other quantity in that list directly from it, using equations in the second list as examples - how in the world does one make sense of all this? Where should one start? What am I missing? I can't even begin with all this there's so much going on :(
Is there not some standard way to begin with something & derive everything from it in an obvious way, & a way to remember all the equations - or a good reasong why you shouldn't care about remembering their names? :(
 A: I would recommend starting by learning how to solve sets of $n$ linear ordinary differential equations (and the "discrete time" analogue: difference equations) with constant coefficients of the form
$$\dot{x}(t)=Ax(t)+f(t),$$
where $x:\mathbb{R}\to\mathbb{R}^n$, $A\in\mathbb{R}^{n\times n}$ and $f:\mathbb{R}\to\mathbb{R}^n$. Then, I'd move on to the case with time-varying coefficients
$$\dot{x}(t)=A(t)x(t)+f(t),$$
where $A:\mathbb{R}\to\mathbb{R}^{n\times n}$. The former is covered in lectures $10-15$ of here while a good resource for the latter are chapter 1-3 of this book (however, it is not free -- maybe someone can suggest one that is?). 
But be warned, there is no silver bullet, the solutions to the general cases will only take you so far--they depend integrals involving $f(t)$ and $A(t)$ and how you (or whether you can at all) evaluate these analytically very much depend on $f(t)$ and $A(t)$ specifically. 
This said, one can always avoid the difficulties of trying to solve them analytically by obtaining instead approximations to the solutions (nowadays, very often, to an arbitrarily precision) by numerically integrating the ODEs on a computer. But I'm starting to go off on a tangent... 
By the way, welcome to MSE.
EDIT: I should mention that you can always re-write a linear $n^{th}$ order differential equation as a set of $n$ first order differential equations as above (but not vice-versa). However, in my experience, it is not much easier to solve the more general set of $n$ differential equations than an $n^{th}$ differential equation and, I think, you'll get more insight doing the general case.
Also, not much changes from considering second order to $n^{th}$ order ODEs when it comes to solving them, so, similarly, I recommend you try directly the general case $n^{th}$ order case (in fact, I think it'll be more understandable). However, these are just my opinions, I'm sure other people think differently.
