Hi, you're welcome. :) I guess it's clear that I suggested a math site so that you might identify some algorithms, which you could bring back to this site, if you need help implementing them.

Side note: I thought "IDE" was a typo. IDE means two things to me in differential equations, integro-differential equation (but your DE has no integral) and implicit differential equation, namely $F(x,y,y')=0$ which is not the form given either. Maybe "f'=f o f" is not f'[x] == f[f[x]]? That is, I'm not sure what the "o" represents, especially in the last example "f'' = f' o". I interpreted the first DE as $f'=f \circ f$, where $\circ$ denotes composition. But perhaps it denotes something else, like convolution? But if so, "f'' = f' o" seems to be missing a right operand for "o". So maybe "o" is not an operation at all. If I was wrong in my interpretation of the DE, then maybe the recommendation to ask on a math site is wrong, too.

Hello all, I reply to myself but it's for Dan,

Daniel (Lichtbau) have you any idea how to do that with Mathematica? (You are of the best developer at Wolftram) If by chance you read this post can you send me a private e-mail telling me if it's doable. Thank you. Kind regards to all,

Jean-Michel

I suggest asking on a mathematics site, especially if you are seeking a symbolic solution. For instance, math.stackexchange.com or mathoverflow.com. An "obvious" solution is $f(0)=0$, but there may be others. I don't know much about this equation.

Here's a series solution approach. If nothing else, it's a start. It requires $f(x_0)=x_0$ for ComposeSeries[f, f] to work, ComposeSeries[f, f] gets unwieldy when the order of the approximation gets high enough.

func[{x0_, n_}, x_] := Block[{a}, a[x0] = x0; Series[a[x], {x, x0, n}]]

With[{x0 = 1},
 coeffrules = Nest[
   With[{f = func[{x0, Length@#}, x]},
     Join[
      #1,
      First@Solve[
        CoefficientList[D[f, x] == ComposeSeries[f, f] /. #1, x],
        Derivative[Length@#][a][x0]]
      ]
     ] &,
   {a[x0] -> x0},
   5
   ];
 res = func[{x0, Length@coeffrules - 1}, x] /. coeffrules
 ]
(*
1 + (x - 1) + 1/2 (x - 1)^2 + 1/3 (x - 1)^3 + 7/24 (x - 1)^4 + 37/120 (x - 1)^5  + O[x - 1]^6
*)

D[res, x]
ComposeSeries[res, res]
(*
1 + (x - 1) + (x - 1)^2 + 7/6 (x - 1)^3 + 37/24 (x - 1)^4 + O[x - 1]^5
1 + (x - 1) + (x - 1)^2 + 7/6 (x - 1)^3 + 37/24 (x - 1)^4 + 269/120 (x - 1)^5  + O[x - 1]^6
*)