From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: --> Looking for Solution to Unusual Function
Date: 21 Mar 1995 16:22:53 GMT

In article <noringD5rErC.JKB@netcom.com>, Jon Noring <noring@netcom.com> wrote:
>As I was toying with some equations today for no other reason than
>recreational "doodling", I wrote down a differential equation, which [to me]
>is quite interesting

With some reservations I might suggest that equations which you write down
just for the heck of it are not as likely to be "interesting" as those 
suggested by some physical or mathematical problem or which are analogous to
equations already studied.

>The problem is to determine f(x) which fulfills the following differential
>equation:
>
>f'(x)= f(f(x))
>
>Is there an analytical solution to this differential equation?  

Well, you could try  f(x) = K x^n; you'll discover that there _is_ a
solution for a certain pair of constants  K  and  n, but they are complex.
You didn't specify what you wanted the domain of  f  to be, so perhaps
this will "count". (Based on this example, I tried conjugating by exp:
let  f(x) = exp(g(log(x))); then we need  g'(x) = exp( g(g(x)) - g(x) + x ).
But I saw no solutions other than  g(x) = linear.)

>Does this equation even have a solution, or is f(x) even continuous?

When you write an ODE, that implies  f  is differentiable, which in turn
forces  f  to be continuous. Of course, you might ask if  f  has a continuous
extension beyond a region on which the  ODE holds, but again since no
domain was given I figure I'm off the hook.

dave

PS There are "trivial" solutions too: take  f  to be locally constant, and
such that all the values in the range of  f  lie in  f^{-1}(0). For
example, f(x) = { 0 if x < 0, and  -1 if x > 0 }