Singularities of Limiting Directions of Generic Higher Order Implicit ODEs

An implicit differential equation of order $n$ is defined as a zero level of a smooth function on the $(n+2)$-dimensional space with a two-dimensional distribution which is the result of natural Goursat prolongation procedure from a standard contact structure in the space of directions on the plane. The solution of this equation is an immersed curve which lies in this level and is tangent to this distribution. Generic metamorphoses of cones of possible directions on the plane of all solutions are classified. This classification is closely related to the classification of generic singularities of first-order implicit differential equations on the plane and to the classification of generic singularities of limiting direction fields of dynamic inequalities on surfaces.