TITLE: What justification can you give for the fact that "most ODEs do not have an explicit solution"?
QUESTION [14 upvotes]: What justification can you give for the fact that "most ODEs do not have an explicit solution"?

REPLY [10 votes]: On the non-linear side, there is a theorem of Hudai-Verenov MR0147699, which says that for
a generic (open dense set) equation $y'=R(x,y)$ the graph of a generic solution is dense.
See also Ilyashenko, MR0247176. This implies that this equations do not have first integrals, so they are not "solvable" in the sense that we teach in the elementary courses.
This is an example of a general theorem on the subject. But of course it is well known from the last 300 years of experience with differential equations. For example, all integrable cases of the rotating top in uniform gravity field are explicitly known, and they are exceptional. There are many other examples like this.