Об алгебраических интегралах дифференциального уравнения

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 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 = {1, 2, … } exists in which = + (). We prove that a contracting derivation of a polynomial ring admits a general integral only if among the indices 1, 2, … 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.

Редакторы
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Издательство
Федеральное государственное автономное образовательное учреждение высшего образования Российский университет дружбы народов (РУДН)
Номер выпуска
2
Язык
Английский
Страницы
105-123
Статус
Опубликовано
Подразделение
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Ссылка
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DOI
-
Номер
-
Том
27
Год
2019
Организации
  • 1 Российский университет дружбы народов
Ключевые слова
Darboux polynomials; algebraic integrals of differential equations; Finite solution; sage; SageMath; maple
Дата создания
20.02.2020
Дата изменения
20.02.2020
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/62430/