# Show that solutions cannot leave the interval.

Consider a first order autonomous equation in $R^1$ with $f(x)$ Lipschitz. Assume x˙=f(x) and Suppose $f(0)=f(1)=0$. I need to show that solutions starting in [0,1] cannot leave this interval. Also I need to find the maximal interval of definition $(T_-,T_+)$ for solutions starting in $[0,1]$. Does such a solution have a limit as $t → T_{+_-}$

Any help is greatly appreciated.

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You should maybe say what your ODE is, I assume you are looking at $\dot{x}=f(x)$. Hint: The constant functions $x_0(t) \equiv 0$ and $x_1(t) \equiv 0$ are solutions, solutions are unique, and so their graphs can never cross each other. – Lukas Geyer Nov 2 '12 at 4:59
@ Lucas what conditions do I have that assure unicity of my solution? How do I show it? – Klara Nov 2 '12 at 5:30
Lipschitz continuity assures uniqueness of solutions. – Lukas Geyer Nov 2 '12 at 5:33
Yes, the interval of existence is the whole real line, and yes, they have limits as $x\to\pm\infty$, since any solution is monotone and bounded. – Lukas Geyer Nov 8 '12 at 22:33
This is a general property of solutions to ODEs of this form. If the solution is not monotone, there has to be a point where $x'(t_0)=0$, so $f(x(t_0))=0$, but then the solution is constant, $x(t) \equiv x(t_0)$, by uniqueness of solutions. – Lukas Geyer Nov 13 '12 at 15:51