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

Abstract

We show that the system x dot = y dot = 2n+2xy+y 2 has the algebraic solution h( x, y) = H n ( x) y + 2 nH 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 ϵ N .