What justification can you give for the fact that "most ODEs do not have an explicit solution"? 



What justification can you give for the fact that "most ODEs do not have an explicit solution"?
If the ODE is linear --and the notion of «explicit» refers to Liouvillian solutions (towers of iterated quadrature and exponential of meromorphic functions)-- then its differential Galois group ([Picard-Vessiot theory](https://mathoverflow.net/questions/140849/solution-of-linear-ode/140851#140851)) must be a solvable algebraic subgroup of $GL\_n (\mathbb C)$. Such subgroups are rare: they define a proper algebraic subvariety. The defining equations correspond to trivial commutations relations, *e.g* $[G,G]=0$ in the Abelian case, encoding the tower of normal subgroups intervening in the definition of [solvable group](http://en.wikipedia.org/wiki/Solvable_group).


In the non-linear context the intuition is the same: nice «transverse structures» are rare.


**Edit**: for a statement regarding the non-linear case, see my paper <http://fr.arxiv.org/abs/1308.6371v2>, Corollary C.


