The structure of general solutions and integrability conditions for rational first-order ODE's

Abstract

In present paper we propose an approach based on examination of the structure of the general solution of equations of the type dy/dx=P(x,y)/Q(x,y), with P and Q polynomials only in y. Under the term structure we mean the dependency character of solution from arbitrary constant. We describe a common form of the structures for foregoing equations. In such a way one can obtain a differential-algebraic polynomial system for undetermined parameters of the structures. The successful solution of this system automatically leads to finding the general solution of ODE's. We demonstrate on examples that proposed method gives, as a first step, the systematic way for obtaining new integrability conditions and general solutions for various families of rational first-order ODE's.