+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: Solutions to f ' (x) = f(f(x)) ? (/showthread.php?tid=804)
Solutions to f ' (x) = f(f(x)) ? - tommy1729 - 08/11/2013
We all know f ' (x) = f (x) for f(x) := C exp(x).
But what is the general solution to f ' (x) = f(f(x)) ?
Of course f(x) = id(0) will do.
I was thinking about its fixpoints : y = f ' (y) = f(f(y))
If y is also a fixpoint of f then
f '' (y) = f ' ( f(y) ) * f ' (y) = [f ' (y)] ^2 = y^2
If y is also a fixpoint of f '' (y) then y^2 = y !
Now since f ' = f(f) we get f^[3](y) * f^[2](y) = y^2
Let f ' (x) = f(f(x)) then f '' (x) = f ' (f(x)) * f ' (x).
Since f ' (x) = f(f(x)) we get f '' (x) = f^[3](x) * f ' (x).
Rearranging gives f '' (x)/f ' (x) = f^[3](x).
Integrating gives ln( f ' (x) ) = integral f^[3](x) dx + C
I feel like Im getting closer to a solution but Im stuck now.
How about subtitution f^[-1](z) = x ?
Anyways if we plug in f(x) = a x^b we get a b x^(b-1) = a (a x^b)^b = a a^b x^(b^2) = a^(b+1) x^(b^2)
Hence we get the system a b = a^(b+1) and b-1 = b^2.
b = (-1)^(1/3) or -(-1)^(2/3) ( yes can be simplified )
ab = a^(b+1) => b = a^b => a = b^(1/b) ( typical tetration )
however what about other solutions ??
regards
tommy1729
RE: Solutions to f ' (x) = f(f(x)) ? - tommy1729 - 08/12/2013
Notice how the superfunction of a x^b involves the generalizations.
e.g. f ' (x) = f^[2,5](x)
regards
tommy1729