Dealing with rational second order ordinary differential equations where both Darboux and Lie find it difficult: The S-function method
TLDR
A new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations, an alternative to the Darbouxian and symmetry approaches, which can succeed in many cases where these two approaches fail.About:
This article is published in Computer Physics Communications.The article was published on 2019-01-01 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Ordinary differential equation & Symbolic computation.read more
Citations
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An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method
TL;DR: In this article, a method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) by generalizing Prelle-Singer has been developed.
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A generalization of the S-function method applied to a Duffing–Van der Pol forced oscillator
TL;DR: In this paper, a generalized S -function method for rational second-order ordinary differential equations (2ODEs) has been proposed, which is able to deal with a class of 2ODEs that are resistant to canonical Lie methods and to Darbouxian approaches.
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First integrals and exact solutions of a class of nonlinear systems
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Solving 1ODEs with functions
TL;DR: A new approach to deal with first order ordinary differential equations (1ODEs), presenting functions is presented, an alternative to the one presented in [1], and the theoretical background to deal, in the extended Prelle-Singer approach context, with systems of 1ODEs is established.
A New S-Function Method searching for First Order Differential Integrals: Faster, Broader, Better
TL;DR: In this paper , the S-function was used to find an integrating factor of a set of first order rational ordinary differential equations (rational 1ODEs) which is shared by the original 2ODE.
References
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Book
Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Book
Symmetry and Integration Methods for Differential Equations
George W. Bluman,Stephen C. Anco +1 more
TL;DR: In this paper, Lie Groups of Transformations and Infinitesimal Transformations (LGTL) are used for dimensionality analysis, modeling, and invariance in Dimensional Analysis, Modeling and Invariance.
Book
Elementary Lie Group Analysis and Ordinary Differential Equations
TL;DR: In this paper, the authors present a Lie Group Analysis of Ordinary Differential Equations (ODE) for the first order and second order differential equations, respectively, and integrate them into Third Order Equations.
Journal ArticleDOI
Elementary first integrals of differential equations
M. J. Prelle,Michael F. Singer +1 more
TL;DR: In this paper, it is shown that it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals, which allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.
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Liouvillian first integrals of differential equations
TL;DR: In this article, it was shown that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a LIOUVILLIAN function of several variables vanishing on the curve defined by this solution, then the system has a nonconstant LIOUVM function that is constant on solution curves in some nonempty open set.