There is yet another direction in which one could take the question, one that makes more contact with the linear and affine structures that one observes in the standard treatment of linear equations (both homogeneous and inhomogeneous). For example, consider the Riccati equation
$$
u'(t) = a_0(t) + a_1(t)u(t) + a_2(t)u(t)^2,\tag{1}
$$
which is nonlinear (when $a_2$ is not zero). A standard fact that is proved in ODE courses is that there is a 'trick' for 'linearizing' this equation, yielding the result that the general solution can be written in the form
$$
u(t) = \frac{c_1b_1(t)+c_2b_2(t)}{c_1b_3(t)+c_2b_4(t)}
$$
for some functions $b_i(t)$ (defined on the entire interval on which the $a_i$ are defined) and constants $c_1$ and $c_2$ (not both zero). Obviously, only the ratio of $c_1$ to $c_2$ matters for specifying $u$, so the ($1$-dimensional) space of solutions to $(1)$ is 'naturally' diffeomorphic to the real projective line $\mathbb{RP}^1$. Note that this representation works even when the solution $u$ has a 'pole'. (For example, consider the Riccati equation $u' = 1 + u^2$, which has $u(t) = \tan t$ as a solution.)

Sophus Lie developed a theory that associated a type of ODE to each homogeneous space of what we now call Lie groups, one that generalized the theory of linear ODE (both homogeneous and inhomogeneous) and Riccati equations in their various forms (which correspond to vector spaces, affine spaces, and projective spaces, thought of as homogeneous spaces of appropriate Lie groups).

It's possible that the OP is interested in learning more about the theory of ODE (and PDE) of Lie type, which is what this theory developed into. For this theory, putting a (possibly non-Hausdorff) manifold structure on the space of solutions is a necessary first step, but it is only the beginning.