As Robert Bryant observed, something like the solubility of $F$ for $u^{(n)}$ needs to be assumed. Then by the trick (due I believe to D'Alembert) of setting $x=(t,u,\dots,u^{(n-1)})$ and dt/ds=1, the equation can always be rewritten
$$
\frac{dx}{ds}=f(x).
$$
This is what most geometers would call the "standard ODE", wherein $f$ is a smooth vector field on the manifold where $x$ evolves.

In this setting, *the space of (maximal connected) solution curves is indeed always a (not necessarily Hausdorff) manifold*. I don't know who first wrote this, but according to this recent [obituary][1] a contender would be J.-M. Souriau in his 1970 book "[Structure des systèmes dynamiques][2]". Translation of the relevant passage:

> **The manifold of motions of a system**
>
> Jean-Marie Souriau has observed that the set of maximal solutions of a differential system on a differentiable manifold possesses itself a natural structure of differentiable manifold, not always separated: thus one can speak of the system's *manifold of motions*. This very simple property is rarely mentioned in the usual courses on differential calculus.  


  [1]: http://smf4.emath.fr/Publications/Gazette/2012/133/smf_gazette_133_97-102.pdf
  [2]: http://books.google.co.uk/books/about/Structure_of_dynamical_systems.html?id=4tBrbryIKQAC