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.
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.

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?

In short: What can we say about the set of all solutions of an ordinary differential equation (ODE) when we there is no uniqueness?

Consider the ODE

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

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

Two fundamental theorems are:

1. If $f\in C(D)$, then there exists a neighbourhood of $x_0$ for which there is a solution $y(x) $ Peano.
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.

The natural question 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?

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.
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.

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?

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.
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.

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?