Classifications of completely integrable implicit second order ordinary differential equations

Journal of Singularities
volume 10 (2014), 271-285

Received 19 December 2012. Received in revised form 17 August 2013.

DOI: 10.5427/jsing.2014.10s

Add a reference to this article to your citeulike library.

Abstract:

An implicit second order ordinary differential equation is said to be completely integrable if there exists at least locally an immersive two-parameter family of geometric solutions on the equation hypersurface like as in the case of explicit equations. An implicit equation may have an immersive one-parameter family of geometric solutions (or, singular solutions) and a geometric solution (or, an isolated singular solution). In this paper, we give a classification of types of completely integrable implicit second order ordinary differential equations and give existence conditions for such families of solutions.

Keywords:

implicit ordinary differential equation, geometric solution, singular solution, complete solution, Clairaut type, reduced type

Mathematical Subject Classification:

Primary 34A26; Secondary 34A09, 34C05, 65L05

Author(s) information:

Masatomo Takahashi
Muroran Institute of Technology
Muroran 050-8585, JAPAN
email: masatomo@mmm.muroran-it.ac.jp