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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-9,
author = {Masatomo Takahashi},
title = {On complete solutions and complete singular solutions of second order ordinary differential equations},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {271-285},
zbl = {1126.34005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-9}
}
Masatomo Takahashi. On complete solutions and complete singular solutions of second order ordinary differential equations. Colloquium Mathematicae, Tome 107 (2007) pp. 271-285. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-9/