On algebraic integrals of a differential equation
We consider the problem of integrating a given differential equation in algebraic functions, which arose together with the integral calculus, but still is not completely resolved in finite form. The difficulties that modern systems of computer algebra face in solving it are examined using Maple as a...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Peoples’ Friendship University of Russia (RUDN University)
2019-12-01
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| Series: | Discrete and Continuous Models and Applied Computational Science |
| Subjects: | |
| Online Access: | http://journals.rudn.ru/miph/article/viewFile/22202/17418 |
| _version_ | 1818215941679349760 |
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| author | Mikhail D Malykh Leonid A Sevastianov Yu Ying |
| author_facet | Mikhail D Malykh Leonid A Sevastianov Yu Ying |
| author_sort | Mikhail D Malykh |
| collection | DOAJ |
| description | We consider the problem of integrating a given differential equation in algebraic functions, which arose together with the integral calculus, but still is not completely resolved in finite form. The difficulties that modern systems of computer algebra face in solving it are examined using Maple as an example. Its solution according to the method of Lagutinskis determinants and its implementation in the form of a Sagemath package are presented. Necessary conditions for the existence of an integral of contracting derivation are given. A derivation of the ring will be called contracting, if such basis B= {m1, m2, } exists in which Dmi= cimi+o (mi). We prove that a contracting derivation of a polynomial ring admits a general integral only if among the indices c1, c2, there are equal ones. This theorem is convenient for applying to the problem of finding an algebraic integral of Briot-Bouquet equation and differential equations with symbolic parameters. A number of necessary criteria for the existence of an integral are obtained, including those for differential equations of the Briot and Bouquet. New necessary conditions for the existence of a rational integral concerning a fixed singular point are given and realized in Sage. |
| first_indexed | 2024-12-12T06:44:05Z |
| format | Article |
| id | doaj.art-1e2b7db4784a4368b4a06f300c34988b |
| institution | Directory Open Access Journal |
| issn | 2658-4670 2658-7149 |
| language | English |
| last_indexed | 2024-12-12T06:44:05Z |
| publishDate | 2019-12-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | Discrete and Continuous Models and Applied Computational Science |
| spelling | doaj.art-1e2b7db4784a4368b4a06f300c34988b2022-12-22T00:34:14ZengPeoples’ Friendship University of Russia (RUDN University)Discrete and Continuous Models and Applied Computational Science2658-46702658-71492019-12-0127210512310.22363/2658-4670-2019-27-2-105-12318090On algebraic integrals of a differential equationMikhail D Malykh0Leonid A Sevastianov1Yu Ying2Peoples’ Friendship University of RussiaPeoples’ Friendship University of RussiaPeoples’ Friendship University of RussiaWe consider the problem of integrating a given differential equation in algebraic functions, which arose together with the integral calculus, but still is not completely resolved in finite form. The difficulties that modern systems of computer algebra face in solving it are examined using Maple as an example. Its solution according to the method of Lagutinskis determinants and its implementation in the form of a Sagemath package are presented. Necessary conditions for the existence of an integral of contracting derivation are given. A derivation of the ring will be called contracting, if such basis B= {m1, m2, } exists in which Dmi= cimi+o (mi). We prove that a contracting derivation of a polynomial ring admits a general integral only if among the indices c1, c2, there are equal ones. This theorem is convenient for applying to the problem of finding an algebraic integral of Briot-Bouquet equation and differential equations with symbolic parameters. A number of necessary criteria for the existence of an integral are obtained, including those for differential equations of the Briot and Bouquet. New necessary conditions for the existence of a rational integral concerning a fixed singular point are given and realized in Sage.http://journals.rudn.ru/miph/article/viewFile/22202/17418darboux polynomialsalgebraic integrals of differential equationsfinite solutionsagesagemathmaple |
| spellingShingle | Mikhail D Malykh Leonid A Sevastianov Yu Ying On algebraic integrals of a differential equation Discrete and Continuous Models and Applied Computational Science darboux polynomials algebraic integrals of differential equations finite solution sage sagemath maple |
| title | On algebraic integrals of a differential equation |
| title_full | On algebraic integrals of a differential equation |
| title_fullStr | On algebraic integrals of a differential equation |
| title_full_unstemmed | On algebraic integrals of a differential equation |
| title_short | On algebraic integrals of a differential equation |
| title_sort | on algebraic integrals of a differential equation |
| topic | darboux polynomials algebraic integrals of differential equations finite solution sage sagemath maple |
| url | http://journals.rudn.ru/miph/article/viewFile/22202/17418 |
| work_keys_str_mv | AT mikhaildmalykh onalgebraicintegralsofadifferentialequation AT leonidasevastianov onalgebraicintegralsofadifferentialequation AT yuying onalgebraicintegralsofadifferentialequation |