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arXiv · 2210.07537

A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations

R. Mohanasubha·M. Senthilvelan
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Abstract

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.

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R. Mohanasubha, M. Senthilvelan. 2022-10-14. A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations. https://doi.org/10.1007/s12043-022-02507-8

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