It is often said that most differential equations (with a solution) do not have an analytic solution, i.e. expressible as a finite combination of elementary functions. Is this proven?
A more formal statement would be: Given the set $S$ of all ODEs that are a finite combination of elementary functions (of fixed order, if that makes the proof simple), the set of ODEs with analytic solution $A$ is smaller than $S$.
How would one measure size in this case? Can $A$ be a dense subset of $S$ in some topology? $A$ could have measure zero and $S$ measure one?
The problem of the symbolic integrability of functions is settled by the Liouville's theorem. The modern developments related to this theory les to the Risch algorithm, which is used in the CAS for automatic intergation.
Some ODE's are solved by one or more ordinary integrations (f.i. first order linear equations with non-constant coefficients), and Liouville's theorem applies. For others, maybe the theorem can be generalized. I wouldn't believe that they are "easier".
Also think that ordinary equations are a trivial case of ODE's, and I don't think we even have a theory of solvability in the non-linear case.
E.g. $$ye^y=x$$ doesn't have an analytical solution in $y$, while $$3e^{3y}+e^y=x$$ does.