On combinatorial differential equations

Abstract We analyse the solution set of first-order initial value differential problems of the form dy dx = ƒ(x, y), y(0) = 0 in the context of combinatorial species in the sense of A. Joyal ( Adv. in Math. 42 (1981), 1–82). It turns out that the situation is much richer than in the case of formal power series: many non-isomorphic combinatorial solutions are possible for a given problem, although they all have the same underlying generating series. We give many examples of this phenomenon and also elaborate a combinatorial Newton-Raphson iterative scheme for the construction of the solutions. The multidimensional case is treated explicitly.

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