Duality and implicit differential equations

We prove some duality results concerning various types of implicit differential equations where F is a smooth function. We show, for instance, that the well folded singularities are self-dual. The results are used to deduce some geometric properties of surfaces in 3-space. So, for example, there are three flecnodal curves at an elliptic flat umbilic and one at a hyperbolic flat umbilic. These curves are tangent to the separatrices of the binary differential equation determining the asymptotic lines.