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How to solve non-linear differential equation $$y'(x) = y(y(x)), \quad y\colon\mathbb{R}\to\mathbb{R}?$$

Of course, $y(x)\not\equiv 0$. If we substitute $y(x) = Ax^n$, we get complex $n$ and $A$. Any numerical solution doesn't work because we can't calculate $y(y(x))$. Series expansion for $y(x)$ is also failed.

Is there a theory of such equations? Numerical solution is not what I want.

kjetil b halvorsen
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Michael Galuza
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  • Well, $y=0$ is one solution. An interesting problem, I must say. I have no idea how to solve it, though. – 5xum Jun 03 '15 at 09:47
  • @5xum, yes, $y(x)\equiv 0$ is a solution, but it's not interesting – Michael Galuza Jun 03 '15 at 09:49
  • Did you find any other solution? – 5xum Jun 03 '15 at 09:49
  • @5xum, only complex. I've tried find $y(x)$ in form exp's, rational functions, but it's derivative and $y(y(x))$ has different forms. – Michael Galuza Jun 03 '15 at 09:52
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    @Martigan, hmm, you are right. But I wish find real and nontrivial solutions – Michael Galuza Jun 03 '15 at 10:19
  • There's a lnk here (whichi s also from that old post) which seems to solve the problem (not finding an explicit $y$ though) http://www.artofproblemsolving.com/community/c7h321705 –  Jun 03 '15 at 10:25
  • @John, I read this post. This equation transformed to system of non-linear DE's, and solution is numerical. I wish to find any non-numerical solution. Or may be there is a theory of such equations, because iterations $f(f(x))$ are popular subject. – Michael Galuza Jun 03 '15 at 10:29
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    Picky question: is this technically a non-linear equation? – Paul Jun 03 '15 at 13:20
  • @Paul, it's a good question. If "non-linear" is all except linear, that's non-linear DE. But there are no expressions like $y(y(x))$ in definition of DE. Therefore, technically it's not DE) – Michael Galuza Jun 03 '15 at 13:25
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    Its a functional-differential equation. Have a look at http://www.mathfile.net/hicstat_fde.pdf (and google "functional-differential equation") – kjetil b halvorsen Jun 03 '15 at 13:29
  • Have you tried working with a power series solution? There are some interesting characteristics when I started along this path. – Paul Jun 03 '15 at 13:33
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    @kjetilbhalvorsen, I'm not sure, but in a functional-differential equation $y'(t) = f(t,y(t),y(u(t)))$ we have function $f$, not $y$. Eqs. like $y'(t) = y(t-1)$ trivially solved by $y(t)=A\exp(\lambda t)$. – Michael Galuza Jun 03 '15 at 13:34
  • @Paul, I've tried, but result is a non-linear infinity system of eqs. – Michael Galuza Jun 03 '15 at 13:35

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