Synthetic solution manifolds for differential equations

AbstractThe space of all solutions for a first-order differential equation can be regarded as a manifold, provided we generalize the traditional notion of differential manifold. We consider two such generalizations using C∞-rings and the smooth Basel topos B. Our definition enables us to define non-standard solutions such as probabilistic ones. There is a sense in which all first-order differential equations have global solutions (possibly non-standard) satisfying given initial conditions. We also prove change of variable theorems and discuss a smoothness condition