On the structure of some typical singularities for implicit ordinary differential equations

TL;DR

This paper investigates the structure of singularities in implicit ordinary differential equations, analyzing their solutions near singular points and classifying various types relevant to physical applications.

Contribution

It provides a detailed analysis of singular solutions and classifies different singularity types in implicit ODEs with minimal additional conditions.

Findings

Classification of singularities such as 'turning' and 'intersection' types.

Explicit description of solutions near singular points.

Identification of conditions leading to different singularity types.

Abstract

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives reduces to the singular surface where the theorem of uniqueness and existence is violated.A set of the singular solution is obtained for the case requiring the minimal number of additional conditions. The various types of singularities correspond to these cases. These ``turning'', ``intersection'' singularities appear in several physical applications.