An improvement to Darboux integrability theorem for systems having a center
Let \(\dot x= P(x,y)\), \(\dot y= Q(x, y)\) be a plane polynomial differential system \((S)\) of degree \(m\). Darboux proved that, if \((S)\) has \({m(m+1)\over 2}\) algebraic solutions, then it has a first integral or an integrating factor of the type \(f^{\lambda_1}_1\cdots f^{\lambda_q}_q\), \(f_i\in \mathbb{C}[x, y]\), \(\lambda_i\in \mathbb{C}\). Such functions are called Darboux first integrals, or, respectively, Darboux integrating factors. During the last decade, Darboux' result was extended in several ways. Here, the following theorem is proved: Let \((S)\) have a center. If \((S)\) has \({m(m+1)\over 2}-[{m+1\over 2}]\) algebraic solutions or exponential factors, then \((S)\) has a Darboux integrating factor.
scientific article; zbMATH DE number 1428534