Area of Interest

I am working on solving rational general solutions of parametrizable ODEs of order 1.

Ordinary differential equations (ODEs) come in two varieties: linear and non-linear. Of course, linear ODEs are just a special case of non-linear ones. However, the solutions of linear ODEs have a linear structure and they can be studied by linear algebra tools. Solutions of non-linear ODEs are more difficult to study and there is no general method for all those differential equations. While one can deal with linear ODEs of any order, the order of a non-linear ODE plays an essential role.

My advisor and I have been interested in determining rational general solutions of parametrizable ODEs of order 1. I.e., we consider an algebraic ODE of the form $F(x,y,y')=0$, where F is a trivariate polynomial and the algebraic surface $F(x,y,z)=0$ admits a rational parametrization ${\cal{P}}(s,t)$. Having a rational parametrization of $F(x,y,z)=0$ allows us to associate the differential equation $F(x,y,y')=0$ with a planar system of autonomous ODEs of order 1 and of degree 1 in the derivatives of the two parameters s and t. We are able to compute a rational general solution of this associated system, so we obtain a rational general solution of $F(x,y,y')=0$ via the parametrization mapping. This geometric approach has first been successfully applied by R. Feng and X-S. Gao in the case of autonomous ODEs and we have been able to extend this approach to the case of non-autonomous parametrizable ODEs of order 1.
Example

Conferences	Talks
First Workshop on Differential Equations by Algebraic Methods (DEAM), February 6-8, 2009, Hagenberg, Austria	Rational general solutions of first order non-autonomous parametric ODEs
Effective Methods in Algebraic Geometry (MEGA'09), June 15-19, 2009, Barcelona, spain	Rational general solutions of first order non-autonomous parametric ODEs
16th International Conference Applications of Computer Algebra (ACA'10), June 24-27, Vlore, Albania	Invariant algebraic curves of rational vector fields and their explicit rational solutions
XII Encuentro de A'lgebra Computational y Aplicaciones (EACA), July 19-21, 2010, Santiago de Compostela, spain	Finding rational solutions of rational systems of autonomous ODEs
Fourth International Workshop on Differential Algebra and Related Topics (DART IV), Beijing, China	Poster: Algebraic Curves and Differential Equations
Second Workshop on Differential Equations by Algebraic Methods (DEAM2), February 9-11, 2011, Linz, Austria	Solving parametrizable ODEs