The ODE f'(x)=f(f(x)), also written as y'=y°y, far surpasses the level at which I learnt the theory and paractice of ODEs, and I am not even sure it has a solution (that is non-trivial). This is not a problem I have encountered anywhere, but rather a thought that is bothering me for quite a while. I have tried proving a non-trivial solution for such a problem does not exist, using various methods (mainly using Picard-Lindëlof theorem in various ways), but in vain. I have also tried finding such a solution, with similiar success.
So, how do I even handle such a problem? Is there a non-trivial solution? And if so, what is the solution? Are there any physical interpretations to such an ODE?
Thank you very much!
