1 Some differential equations in SDG Anders Kock and Gonzalo

We intend to comment on some of those aspects of the theory of differential equations which we think are clarified (for us, at least) by means of the synthetic method. By this, we understand that the objects under consideration are seen as objects in one sufficiently rich category (model for SDG), allowing us, for instance, to work with nilpotent numbers, say d ∈ R with d = 0; but the setting should also permit the formation of function spaces, so that some of the methods of functional analysis, become available, in particular, the theory of distributions. The specific topics we treat are generalities on vector fields and the solutions of corresponding firstand second-order ordinary differential equations; and also some partial differential equations, which can be seen in this light, the waveand heat-equation on some simple spaces, like the line R. For these equations, distribution theory is not just a tool, but is rather the essence of the matter, since what develops through time, is a distribution (of heat, say), which, as stressed by Lawvere, is an extensive quantity, and as such behaves covariantly, unlike density functions (which behave contravariantly); and the distributions may have no density function, in particular in the setting of model for SDG where all functions are smooth. When we consider these partial differential equations, we shall follow an old practice and sometimes denote derivative d/dt with respect to “time” by a dot, ḟ , whereas differential operators with respect to space variables are denoted ∂f/∂x, f ′, ∆(f), etc.