TITLE: Proof that "many differential equations cannot be solved using analysis"?
QUESTION [7 upvotes]: The wiki page on Numerical methods for ordinary differential equations states that "many differential equations cannot be solved using analysis." Is this literally true, or do they mean to say "it is not known how to solve many differential equations using analysis"? If it is literally true, how is that provable?

REPLY [3 votes]: They literally mean that there does not exist a symbolic expression in terms of "elementary" functions.
For the simplest differential equation
$$ y' = f(x) $$
it's clear that the solution for $y$ is simply the antiderivative of $f$. For some elementary functions $f$, it is possible to prove that there does not exist any elementary function whose derivative is $f$.
I have never studied the topic myself, but I know of several leads one could follow to study the subject:

Risch's algorithm - an algorithm for finding elementary antiderivatives or determining whether one does not exist
Liouville's theorem - speaks of a property that derivatives of elementary functions must have
Differential Galois theory - An analog of Galois theory relevant to the topic

That said, one can always extend the class of functions one considers. e.g. if you are really interested in a specific function $f$ and need to work with its antiderivative, then one gives the antiderivative a name and figures out its properties and how to work with and compute with it.
There is, I believe, a large amount of research (open and proprietary) in this direction; e.g. developing the classes of functions and algorithms used internally by symbolic math packages such as Mathematica.