Numerical solutions for implicit differential equations with singularities

In this paper we introduce a technique to deal with implicit differential equations exhibiting singularities. Our approach is a geometrical one, we use the concept of contact structure on a manifold associated with the differential equation. In this setting we prove an existence and uniqueness theorem. We also show how it relates to known geometric results for this kind of equation. We also indicate how the method can be implemented by using continuation methods techniques and the BDF (Backward Differentiation Formula).