The thing that always bugged me about differential equations when I encountered them in the MIT analog circuits course was that the professor said that the way to figure them out was to use "guesswork". For example, the diffeq describing how RLC circuits work is well known, but can not be solved using a systematic algorithm that a person could do by hand.
You might be confusing straightforward linear systems of differential equations (some exceptions when you have a nasty heterogenous part, but which are 'always' solvable by a standard method) with so-called Differential Algebraic Equations (DAEs).
These beasts often pop up when trying to solve larger electrical circuits with time-dependent elements (ie. RLC). The only thing you can really do to solve them usually is linearizing them approximate them by some integration scheme (this is what Spice, the electronics simulator is based on).
Uni-level math on the reals always bugged me for the exact reason that I thought well-defined problems not involving funny self-reference tricks ought to have algorithms. Despite this, math professors kept presenting us with patterns to match, but no exhaustive algorithms.
Little did I know, considering the reals are uncountable, so you can't really come up with exhaustive algorithms for lots of things dealing with them.
Perhaps you would like to study constructive analysis, a la Bishop? You lose a lot of intuition in the hypotheses, which necessarily become more complicated—or perhaps I should say that you are forced to develop a different intuition about hypotheses—but you gain conclusions of exactly the sort that you want.
EDIT: Also, the fact that a structure is uncountable doesn't prevent you from operating algorithmically on it (not that you claimed it did!); it just means that there are some elements of it that cannot be singled out algorithmically.
> Despite this, math professors kept presenting us with patterns to match, but no exhaustive algorithms.
This is basically what math is all about. If a problem can be effectively and easily solved algorithmically, it is considered trivial, and little attention is given to it. The focus is on actually difficult problems, ones that need clever tricks to get a handle of.
To be frank: bull. A problem that can always be solved algorithmically is "trivial", in the mathematical sense, but since the mind is algorithmic, you can't just say, "Well, clever tricks, not algorithms." A mathematician's mind works somehow.
then it seems like you may be using 'algorithmic' to mean something like "unfolding according to (possibly unknown, possibly probabilistic) laws", in which case it seems so broad a term as to be almost useless.
>then it seems like you may be using 'algorithmic' to mean something like "unfolding according to (possibly unknown, possibly probabilistic) laws"
They're certainly currently unknown, and we have good reason to believe they're probabilistic.
Even such vague, loose descriptions are better than invoking "mathematical intuition" or "it just comes to me" or other explanations for how one does math.
There are ways to systematically get at DE solutions, the first that come to mind are power series or fourier series or decomposing into other orthogonal functions (of course, assuming the solution obeys relevant boundary conditions), but that won't give you an elegant result unless you have the time to wade through the recursion relations and sum it up. Sometimes, a smart guess just gives you the result, so why not do that?
There are methods to solve them. RLC in particular can be approached with Laplace Transforms. However these kinds of equations need certain approximations and simplifications in order to become analytically tractable.
> Isn't there a theorem that says that no closed-form, analytic solutions to such equations can be found, in general?
'Analytic' has a mathematical meaning (https://en.wikipedia.org/wiki/Analytic_function) which is probably not what you mean here. Taking it in the more colloquial (Eulerian) sense of "given by a formula", not only are there such general results, but there are even specific functions for which it is known that no elementary (https://en.wikipedia.org/wiki/Elementary_function) anti-derivative exists (the strong form of "impossible to solve" that I mentioned above), and an algorithm for deciding of a given function whether it has an elementary anti-derivative (https://en.wikipedia.org/wiki/Risch_algorithm).
