A family of non-Darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree

摘要

We show that the system x = 1, y = 2n+2xy+y(2) has the algebraic solution h(x, y) = H-n(x)y + 2nH(n-1)(x), where H-n(x) is the Hermite polynomial of degree n, and the system is not Darboux integrable and has no Darboux integrating factor for any n is an element of N. (C) 2003 Elsevier Science Ltd. All rights reserved. [References: 9]