# Autonomous ODE: non-uniqueness of solutions

1. Jun 20, 2012

### setthino

1. The problem statement, all variables and given/known data
We have the autonomous ODE:
$\dot{x} = f(x), x \in \mathbb{R}$
first we define the following sets:
$E := \{f(x) = 0\}$
$E^+ := \{f(x) > 0\}$
$f(x)$ is continuous so $E$ is a closed set and $E^+$ is an open set.
$x_0 \in (x_-,x_+)$ where $(x_-,x_+)$ is the connected component of $E^+$ that contains $x_0$.
In addition we define the function:
$\Psi : (x_-,x_+) \rightarrow \mathbb{R}$
$\Psi(z) := \int_{x_0}^z \frac{dy}{f(y)}$
Now the texbook says that if:
$lim_{z\to x_{\pm}} \Psi(z) = l^{\pm}$ is finite, we lose the uniqueness of the solutions and this is quite easy to understand, but what comes after is not :(
It says that we have at least 2 solutions of the equation that at the istant $t = l^{\pm}$ coincide at $x^+$; what are these? are there others?
3. The attempt at a solution
I struggled for hours and i think one of the solutions maybe the function:

$x(t) = \begin{cases} x_- & t = l^- \\ \Psi^{-1}(t-t_0) & l^- < t < l^+ \\ x^+ & t = l^+ \end{cases}$

i.e. i extented the solution x(t) = $\Psi^{-1}(t-t_0)$ at the extremes of $(x_-,x_+)$
but i can't think of other solutions