论文标题
一项关于Darboux多项式的研究及其在确定其他集成性量词中的重要性：三阶非线性普通微分方程的案例研究
A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations
论文作者
论文摘要
在本文中，我们提出了一种从darboux多项式来推导延伸的prelle-singer方法的量词的方法，用于三阶非线性普通微分方程。通过了解Darboux多项式及其辅因子，我们在不评估Prelle-Singer方法的确定方程式的情况下提取了扩展的Prelle-Singer方法的数量。我们考虑三种已知的Darboux多项式的情况。在第一种情况下，我们通过利用来自两个已知的Darboux多项式的Prelle-Singer方法的量词来证明给定的三阶非线性方程的整合性。如果我们只知道一个darboux多项式，那么给定方程的集成性将被视为案例$ 2 $。同样，案例$ 3 $讨论了给定系统的集成性，其中我们有两个Darboux多项式和一组Prelle-Singer方法数量。已建立的互连不仅有助于在不求解基础确定方程的情况下得出可集成的量词。它还提供了一种证明完整的集成性的方法，并帮助我们得出给定方程的一般解决方案。我们通过三个不同的示例演示了此过程的实用性。
In this paper, we present a method of deriving extended Prelle-Singer method's quantifiers from Darboux Polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and its cofactors, we extract the extended Prelle-Singer method's quantities without evaluating the Prelle-Singer method's determining equations. We consider three different cases of known Darboux polynomials. In the first case, we prove the integrability of the given third-order nonlinear equation by utilizing the Prelle-Singer method's quantifiers from the two known Darboux polynomials. If we know only one Darboux polynomial, then the integrability of the given equation will be dealt as case $2$. Likewise, case $3$ discuss the integrability of the given system where we have two Darboux polynomials and one set of Prelle-Singer method quantity. The established interconnection not only helps in deriving the integrable quantifiers without solving the underlying determining equations. It also provides a way to prove the complete integrability and helps us in deriving the general solution of the given equation. We demonstrate the utility of this procedure with three different examples.
