Seminar talk, 27 March 2013

Speaker: Pavel Bibikov

Title: On classification of the ordinary differential equations solved for the highest derivative

Abstract:
The problem of classification of differential equations with respect to point and contact transformations is one of the oldest and most important problems in the theory of differential equations. For all that, the most interesting and important objects to study appear to be the ordinary differential equations solved for the highest derivative, i.e., the equations of the form

$(1)\qquad y^{(n)}=f{\bigl (}x,y,y',\ldots ,y^{(n-1)}{\bigr )}$.

First results in this field were obtained by Sophus Lie:

1. ordinary differential equation $(1)$ of the first order in the neighborhood of nonsingular point is point equivalent to the equation $y'=0$;
2. ordinary differential equation $(1)$ of the second order in the neighborhood of nonsingular point is equivalent to the equation $y''=0$.

Thus, what is of interest is the point classification of ordinary differential equations of form $(1)$ for $n\geqslant 2$ and the contact classification for $n\geqslant 3$.

The case $n=2$ for equations of form $(1)$ in general position was studied by Tresse, a student of Sophus Lie. A detailed description of this result was given by Kruglikov.

The talk will discuss contact and point classifications of equations of form $(1)$ in general position for $n\geqslant 3$. We describe sets of differential invariants of these equations and relations between them that completely define classes of contact (point) equivalence given equations of form $(1)$.