Symbolic methods for systems of implicit ordinary differential equations

This contribution deals with equivalence problems for systems of implicit ordinary differential equations. Equivalence means that every solution of the original set of equations is a solution of some normal form, and vice versa. The system is identified with the submanifold in a suitable jet-space, defined by the equations. Therefore, a short introduction to jet theory is presented, as well as its application to systems of differential equations. We present several results for well-determined and under-determined systems and give formulas that describe the transform to an appropriate normal form. Apart from the theoretical results, we give several sketches of computer algebra-based algorithms necessary to solve these problems efficiently.