As is well known, the space of solutions of a linear ODE with, say, $\mathcal{C}^\infty$ coefficients on $\mathbb{R}$ is a finite dimensional affine space (a vector space, in the homogeneus case).



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> What is the structure of the space of solutions of a *non linear* ODE? In which cases does it have a "natural" structure of finite dimensional smooth manifold? 
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For the question to make sense, some conditions on the form of the equation must be put. Let's assume the ODE is of the form


$$F(t,u(t),u'(t),u''(t),\dots,u^{(n)}(t))=0$$


where $F:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}$ is a smooth function. 



What is the structure of the space of solutions of a non linear ODE?
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](http://smf4.emath.fr/Publications/Gazette/2012/133/smf_gazette_133_97-102.pdf) a contender would be J.-M. Souriau in his 1970 book "[Structure des systèmes dynamiques](http://books.google.co.uk/books/about/Structure_of_dynamical_systems.html?id=4tBrbryIKQAC)". Translation of the relevant passage:



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> **The manifold of motions of a system**
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> 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. 
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