  The problem of counting ramified covers of a Riemann surface up to
homeomorphism was proposed by Hurwitz in the late 1800's. This problem
translates combinatorially into factoring a permutation of specified cycle
type, with certain conditions on the cycle types of the factors, such as
minimality and transitivity.
  Goulden and Jackson have given a proof for the number of minimal, transitive
factorizations of a permutation into transpositions. This proof involves a
partial differential equation for the generating series, called the Join-Cut
equation. Recently, Bousquet-M\'{e}lou and Schaeffer have found the number of
minimal, transitive factorizations of a permutation into arbitrary unspecified
factors. This was proved by a purely combinatorial argument, based on a direct
bijection between factorizations and certain objects called $m$-Eulerian trees.
  In this paper, we give a simple partial differential equation for
Bousquet-M\'{e}lou and Schaeffer's generating series, and for Goulden and
Jackson's generating series, as well as a new proof of the result by
Bousquet-M\'{e}lou and Schaeffer. We apply algebraic methods based on
Lagrange's theorem, and combinatorial methods based on a new use of
Bousquet-M\'{e}lou and Schaeffer's $m$-Eulerian trees.
