Rational First Integrals of Separable Differential Equations

In this paper, we present a necessary and sufficient condition for the existence of rational first integrals of the following separable differential equation: dydx=f(x)g(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{dy}{dx}=f(x)g(y) \end{aligned}$$\end{document}where f(x), g(y) are two univariate rational functions. We also present an algorithm to verify the condition and to compute a rational first integral when the condition is satisfied.