Let \(f(t,y,{y^\prime}) = \sum\nolimits_{i = 0}^n {{a_i}(t,y){y^{\prime i}} = 0} \) be an irreducible first order ordinary differential equation with polynomial coefficients. Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t, y, y′) = 0 is of degree not greater than C. Examples show that this degree bound C depends not only on the degrees of f in t, y, y′ but also on the coefficients of f viewed as the polynomial in t, y, y′. In this paper, the authors show that if f satisfies deg(f, y) < deg(f, y′) or
\(\mathop {\max}\limits_{i = 0}^n \{\deg ({a_i},y) - 2(n - i)\} > 0,\)
then the degree bound C only depends on the degrees of f in t, y, y′, and furthermore we present an explicit expression for C in terms of the degrees of f in t, y, y′.
