An algebraic method for quasi-linear first-order ODEs

Abstract

In this paper, we consider the class of quasi-linear first-order ODEs of the form y′ = P(x; y), where P is a polynomial in y with coefficients in ℂ(x), and study their algebraic solutions. Our method is intrinsically based on algebraic function field theory. We give an upper bound for the degrees of algebraic solutions which are in a quadratic extension of ℂ(x).