Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complicated solutions or be unsolvable. Is there some deeper reasoning behind why it is so rare to find solutions? Or is it just that every time we can solve differential equations, it is just an algebraic coincidence?
I reviewed the existence and uniqueness theorems for differential equations and did not find any insight. Nonetheless, perhaps the answer can be found among these?
A huge thanks to anyone willing to help!
Update: I believe I have come up with an answer to this odd problem. It is the bottom voted one just because I posted it about a month after I started thinking about this question and all you’re inputs, but I have taken all the responses on this page into consideration. Thanks everyone!
Let’s consider the following, very simple, differential equation: f′(x)=g(x), where g(x) is some given function. The solution is, of course, f(x)=∫g(x)dx, so for this specific equation the question you’re asking reduces to the question of “which simple functions have simple antiderivatives”. Some famous examples (such as g(x)=e−x2) show that even simple-looking expressions can have antiderivatives that can’t be expressed in such a simple-looking way.
There’s a theorem of Liouville that puts the above into a precise setting: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra). For more general differential equations you might be interested in differential Galois theory.