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A. Kock and G. E. Reyes, Ordinary differential equations and their exponentials, Central European J. of Math. 4 (2006), 64-81

Vector fields or, equivalently, ordinary differential equations have long been considered, heuristically, to be the same as “infinitesimal (pointed) actions” or “infinitesimal flows”, but it is only with the development of Synthetic Differential Geometry (SDG) that we have the tools to formulate these notions and prove their equivalence in a rigourous mathematical way. We exploit this fact to define the exponential of two ordinary differential equations as the exponential of the corresponding infinitesimal actions. The resulting action is seen to be the same as a partial differential equation whose solutions may be obtained by conjugation from the solutions of the differential equations that make up the exponential. Furthermore, we show that this method of conjugation is equivalent, under some conditions, to the method of change of variables, widely used to solve differential equations.

Ordinary differential equations and their exponentials