On complete solutions and complete singular solutions of second order ordinary differential equations

Abstract

Abstract. A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular solution of implicit second order ordinary differential equations.

Classifications of implicit second-order ordinary differential equations of Clairaut type

“…If f is completely integrable, then there exists a foliation on F −1 (0) whose leaves are geometric solutions. We call (µ, ν, f ) an equation with complete integral (see [2,13,14] for details).…”