Timeline for Is there a general solution for the differential equation $f''(x) = f(f(x))$?
Current License: CC BY-SA 4.0
8 events
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| Feb 22, 2021 at 0:27 | comment | added | Robert Bryant | @MichaelEngelhardt: You're welcome. As Will Sawin pointed out in a comment to my answer, though, my speculation that the formal power series solutions might always have a positive radius of convergence was too optimistic; he showed that this cannot be true. | |
| Feb 21, 2021 at 22:22 | comment | added | Michael Engelhardt | @RobertBryant - Ah! I had been wondering whether these two solutions would fit into the general scheme you were developing. And one can make sense of them locally around a whole sequence of fixed points ... nice. Thank you! | |
| Feb 21, 2021 at 19:47 | comment | added | Robert Bryant | It's an interesting fact that these two (multi-valued) solutions are, in fact, special cases of the solutions with fixed points that I mention in my answer. What's interesting is that the fixed points are repelling fixed points, since, at a fixed point $z$ of one of these solutions, one has $|f'(z)|=\sqrt{2}>1$. In particular, this shows that the formal power series solutions around fixed points that I indicate can have a positive radius of convergence, even when the formal fixed point is repelling. Perhaps the formal solutions with fixed point always have a positive radius of convergence? | |
| Feb 18, 2021 at 2:42 | history | edited | user44143 | CC BY-SA 4.0 |
simplified expression for $a$; avoided suggestion that these are the only solutions
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| Feb 17, 2021 at 15:37 | comment | added | Michael Engelhardt | @AlexandreEremenko - I do not know that. I certainly expect there are more. They won't be as simple. | |
| Feb 17, 2021 at 14:53 | comment | added | Alexandre Eremenko | How do you know that there are no other solutions? | |
| Feb 17, 2021 at 8:25 | history | edited | Michael Engelhardt | CC BY-SA 4.0 |
added 3 characters in body
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| Feb 17, 2021 at 5:23 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |