Rational general solutions of first order non-autonomous parametrizable ODEs

Abstract

In this paper we study non-autonomous algebraic ODEs F(x,y,y′)=0, where F(x,y,z)∈Q¯[x,y,z], provided a proper rational parametrization P(s,t) of the corresponding algebraic surface F(x,y,z)=0. We show the relation between a rational general solution of the non-autonomous differential equation F(x,y,y′)=0 and a rational general solution of its associated autonomous system with respect to P(s,t). The degrees of a rational solution (s(x),t(x)) of the associated system are studied by giving a degree bound for t(x) in terms of the degree of s(x) and the degree with respect to s of the first component of P(s,t). We also give a criterion for the existence of rational general solutions of the associated system provided a degree bound of its rational general solutions. The criterion is based on the vanishing of the differential pseudo remainder of Gao’s differential polynomials with respect to the chain of differential polynomials derived from the associated system. We use this criterion to classify all autonomous linear systems of ODEs of order 1 having a rational general solution.