On Liouvillian integrability of the first-order polynomial ordinary differential equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:395

Article

Gine, Jaume¹  Llibre, Jaume² 

[1] Univ Lleida, Dept Matemat, Lleida 25001, Catalonia, Spain

[2] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Catalonia, Spain

关键词: Liouvillian integrability;    Invariant algebraic curve;    Riccati differential equation;    Abel differential equation;   

DOI  :  10.1016/j.jmaa.2012.05.072

来源: Elsevier

【 摘 要 】

Recently, the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves; see Gine and Llibre (2012) [8]. In this note, we prove that, if a complex differential equation of the form y' = a(0)(x) +a(1) (x)y + ... + a(n) (x)y(n), with a(i)(x) polynomials for i = 0, 1,..., n, a(n)(x) not equal 0, and n >= 2, has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n = 1, i.e., for linear polynomial differential equations. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free