On combinatorial differential equations (Q1090823)

This article analyzes the solution set of a first order initial value differential problem of the form \(dy/dx=f(x,y)\), \(y(0)=0\) in the context of combinatorial species in the sense of A. Joyal. It turns out that many non-isomorphic combinatorial solutions are possible for a given problem, although they all have the same underlying generating series. The author gives many examples of this phenomenon and also elaborates a combinatorial Newton-Raphson iterative scheme for the construction of the solutions. The multidimensional case is treated explicitly.