  In 1994, P.-G. Becker and W. Bergweiler listed all the differentially
algebraic solutions of three famous functional equations: the Schr{\"o}der's,
B{\"o}ttcher's and Abel's equations. The proof of this theorem combines various
domains of mathematics. This goes from the theory of iteration, which gave
birth to these equations, to the algebro-differential notion of coherent
families developed by M. Boshernitzan and L. A. Rubel. This survey is an
excursion into the history of these equations, in order to enlighten the
different pieces of mathematics they bring together and how these parts fit
into the result of P.-G. Becker and W. Bergweiler.
