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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2006 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2006
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146113 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862649514589945856 |
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| author | Kosovtsov, Y.N. |
| author_facet | Kosovtsov, Y.N. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-01T15:53:09Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146113 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T15:53:09Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
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| spelling | Kosovtsov, Y.N. 2019-02-07T13:44:52Z 2019-02-07T13:44:52Z 2006 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 https://nasplib.isofts.kiev.ua/handle/123456789/146113 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. I would like to thank the referees for extensive comments and suggestions regarding of earlier versions of this paper and Reece Heineke for a careful reading of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms Article published earlier |
| spellingShingle | Finding Liouvillian First Integrals of Rational ODEs of Any Order in Finite Terms Kosovtsov, Y.N. |
| title | 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_short | 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146113 |
| work_keys_str_mv | AT kosovtsovyn findingliouvillianfirstintegralsofrationalodesofanyorderinfiniteterms |