I have a differential equation given by $ \frac{1}{c^2}=f(\beta)(f'(\beta)^2+1)$, where c ist a positive constant and we have that at some point $\beta'$, we have $f(\beta')=y>0$. Now the question is whether each solution to this ODE is unique, if we do not want to consider solution that are constant on an interval. My problem is the following: obviously there are plenty of solutions to this ODE that describe functions that are somehow constant on some interval. but what is about solutions that are not constant on an interval. Is there more than one solution to this equation? How could I proof this? I mean the problem is, that one cannot write this ODE as a function of f' directly, so theorems like Picard Lindelöf will not apply to this ODE. Unfortunately I do not know many theorems about ODEs so maybe this question is an easy one.

  • $\begingroup$ You don't have the uniqueness of ODE, let alone its solutions... $\endgroup$ – Artem May 17 '13 at 19:26
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    $\begingroup$ sorry, I do not understand your remark, could you say it differently? $\endgroup$ – user66906 May 17 '13 at 19:45
  • $\begingroup$ You have more than one ODE -- actually two: $f'(\beta)=\pm \sqrt{1/(c^2 f(\beta))-1}$. $\endgroup$ – Artem May 17 '13 at 19:46
  • $\begingroup$ so what should this be telling me? $\endgroup$ – user66906 May 17 '13 at 20:27
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    $\begingroup$ It should be telling you that this fact (two ODEs) answers your question about uniqueness (negatively). $\endgroup$ – Artem May 17 '13 at 20:54

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