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Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms

Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms

It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. Th...

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Main Author: Kosovtsov, Y.N.
Format: Article
Language:English
Published: Інститут математики НАН України 2006
Series:Symmetry, Integrability and Geometry: Methods and Applications
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/146113
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spelling irk-123456789-1461132019-02-08T01:23:11Z Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms Kosovtsov, Y.N. It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. These results lead to a partial algorithm for finding Liouvillian first integrals. However, there are two main complications on the way to obtaining polynomials in the integrating factor form. First of all, one has to find an upper bound for the degrees of the polynomials in the product above, an unsolved problem, and then the set of coefficients for each of the polynomials by the computationally-intensive method of undetermined parameters. As a result, this approach was implemented in CAS only for first and relatively simple second order ODEs. We propose an algebraic method for finding polynomials of the integrating factors for rational ODEs of any order, based on examination of the resultants of the polynomials in the numerator and the denominator of the right-hand side of such equation. If both the numerator and the denominator of the right-hand side of such ODE are not constants, the method can determine in finite terms an explicit expression of an integrating factor if the ODE permits integrating factors of the above mentioned form and then the Liouvillian first integral. The tests of this procedure based on the proposed method, implemented in Maple in the case of rational integrating factors, confirm the consistence and efficiency of the method. 2006 Article Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms / Y.N. Kosovtsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 22 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 34A05; 34A34; 34A35 http://dspace.nbuv.gov.ua/handle/123456789/146113 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. These results lead to a partial algorithm for finding Liouvillian first integrals. However, there are two main complications on the way to obtaining polynomials in the integrating factor form. First of all, one has to find an upper bound for the degrees of the polynomials in the product above, an unsolved problem, and then the set of coefficients for each of the polynomials by the computationally-intensive method of undetermined parameters. As a result, this approach was implemented in CAS only for first and relatively simple second order ODEs. We propose an algebraic method for finding polynomials of the integrating factors for rational ODEs of any order, based on examination of the resultants of the polynomials in the numerator and the denominator of the right-hand side of such equation. If both the numerator and the denominator of the right-hand side of such ODE are not constants, the method can determine in finite terms an explicit expression of an integrating factor if the ODE permits integrating factors of the above mentioned form and then the Liouvillian first integral. The tests of this procedure based on the proposed method, implemented in Maple in the case of rational integrating factors, confirm the consistence and efficiency of the method.
format Article
author Kosovtsov, Y.N.
spellingShingle Kosovtsov, Y.N.
Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Kosovtsov, Y.N.
author_sort Kosovtsov, Y.N.
title Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
title_short Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
title_full Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
title_fullStr Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
title_full_unstemmed Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms
title_sort finding liouvillian first integrals of rational odes of any order in finite terms
publisher Інститут математики НАН України
publishDate 2006
url http://dspace.nbuv.gov.ua/handle/123456789/146113
citation_txt Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms / Y.N. Kosovtsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 22 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT kosovtsovyn findingliouvillianfirstintegralsofrationalodesofanyorderinfiniteterms
first_indexed 2023-05-20T17:23:51Z
last_indexed 2023-05-20T17:23:51Z
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