  This paper describes how to define and work with differential equations in
the abstract setting of tangent categories. The key notion is that of a curve
object which is, for differential geometry, the structural analogue of a
natural number object. A curve object is a preinitial object for dynamical
systems; dynamical systems may, in turn, be viewed as determining systems of
differential equations. The unique map from the curve object to a dynamical
system is a solution of the system, and a dynamical system is said to be
complete when for all initial conditions there is a solution. A subtle issue
concerns the question of when a dynamical system is complete, and the paper
provides abstract conditions for this.
  This abstract formulation also allows new perspectives on topics such as
commutative vector fields and flows. In addition, the stronger notion of a
differential curve object, which is the centrepiece of the last section of the
paper, has exponential maps and forms a differential exponential rig. This rig
then, somewhat surprisingly, has an action on every differential object and
bundle in the setting. In this manner, in a very strong sense, such a curve
object plays the role of the real numbers in standard differential geometry.
