PhD Defence: Rational General Solutions of First-Order Algebraic ODEs

Solving algebraic ordinary differential equations (AODEs) has been of interest for a long time and is still an active topic of research. Several classical AODEs have been studied by different methods. It is natural to see them in a bigger class, in which all the equations can be treated uniformly. In this Ph.D. thesis we present an algebraic geometric method to determine the existence of a rational general solution of a parametrizable first-order AODE; if the answer is yes, we will find such a general solution. Our method is based on a rational parametrization of a rational algebraic surface, obtained by neglecting the differential aspect of the differential equation, and on the degree bound of a rational general invariant algebraic curve of the associated system derived from this parametrization. This approach enables us to treat a wide class of parametrizable first-order AODEs in a uni form way.