Abstract
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.
Highlights
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 problem of integrating differential equations in algebraic functions arose as early as the 1630s, when Forimond de Beaune proposed to Descartes several “inverse tangent problems” [1, Pp. 510–518]
The classical problem of finding an algebraic integral has again become relevant in connection with the development of algorithms for the symbolic solution of differential equations suitable for implementation in modern computer algebra systems [7, 8]
Summary
3, Kaiyuan Road, Kaili, 556011, China (received: October 24, 2019; accepted: November 14, 2019). 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. Necessary conditions for the existence of an integral of contracting derivation are given. 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. Key words and phrases: Darboux polynomials; algebraic integrals of differential equations; finite solution; Sage; Sagemath; Maple
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