**In short:** What can we say about the collection of all solutions of an ODE when we don't have uniqueness?


When we teach a first course in ODE's, we look at the equation

   $f:D\to \mathbb{R}, \quad D\subseteq \mathbb{R}^2,$

   $y'(x) = f(x,y),\quad y(x_0 ) = y_0, \quad (x_0,y_0 )\in D $

and prove two theorems

 1. If $f\in C(D)$, then there exists a neighbourhood of $x_0$ for which there is a solution $y(x) $ [Peano Theorem][1].
 2. If $f$ is also Lipschitz in $y$, then there exists a neighbourhood of $x_0$ in which $y(x)$ exist and is a unique solution.[Picard Lindelöf][2].




The natural question, which I tried to "google" and did not find an answer to, is

> What can be generally said about the set of all solutions when there is no uniqueness, i.e. $f$ is continuous but not Lipschitz?




    


  [1]: https://en.wikipedia.org/wiki/Peano_existence_theorem%20%22Peano%20Theorem%22.
  [2]: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem