On algebraic integrals of a differential equation

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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 Lagutinski’s 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.

作者简介

Mikhail Malykh

Peoples’ Friendship University of Russia

编辑信件的主要联系方式.
Email: malykh-md@rudn.ru

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Leonid Sevastianov

Peoples’ Friendship University of Russia

Email: sevastianov-la@rudn.ru

professor, Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Yu Ying

Peoples’ Friendship University of Russia

Email: yingy6165@gmail.com

postgraduate student of Department of Applied Probability and Informatics; assistant professor of Department of Algebra and Geometry

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 3, Kaiyuan Road, Kaili, 556011, China

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