The Method of Finding Algebraic Integral for First-order Differential Equations

Cover Page


Article is devoted to search of algebraic integrals of the ordinary differential equations in the systems of computer algebra. The main attention is paid to development of practical instructions for work with an original package for Sage called in honor of M. N. Lagutinski. At the beginning of article Beaune’s problem is formulated: for a given differential equation, we need to identify whether it is in the form of rational integral, and if the answer is true, we need to quadrature it. The difficulties of finding the upper bound of the integral order and its value for solving differential equations practically are discussed, bounded Beaune’s problem is formulated. Our work is based on the method of M. N. Lagutinski. The theory and its realization are tested on the problems from Text-Book on Differential Equations by A. F. Filippov. The numerical experiments, which were carried out, show that the method makes it possible to identify the existence of the rational integral without taking much resources and time. However, using the method to calculate integrals is very time-consuming. In conclusion, recommendations on the optimal use of the method of Lagutinski are given. All calculations are executed in the computer algebra system Sage.

About the authors

M D Malykh

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

Ying Yu

Peoples’ Friendship University of Russia (RUDN University)

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

graduate student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)


  1. E. Hairer, G. Wanner, C. Lubich, Geometric Numerical Integration. StructurePreserving Algorithms for Ordinary Differential Equations, Springer, Berlin Heidelberg New York, 2000.
  2. R. Descartes, Geometry with the Appendix of Selected Works of P. Fermat and Descartes’ Correspondence, GONTI NKTP SSSR, Moscow-Leningrad, 1938, in Russian.
  3. G. Ch´eze, Computation of Darboux Polynomials and Rational First Integrals with Bounded Degree in Polynomial Time, Journal of Complexity 27 (2011) 246–262. doi: 10.1016/j.jco.2010.10.004.
  4. W. W. Golubev, Vorlesungen u¨ber Differentialgleichungen im Komplexen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.
  5. M. N. Lagutinski, The Application of Polar Operations to Integration of the Ordinary Differential Equations in Finite Terms, Communications of the Kharkov Mathematical Society. The second series 12 (1911) 111–243, in Russian.
  6. M. N. Lagutinski, On Some Polynoms and Their Aplication for Algebraic Integration of Ordinary Differential Algebraic Equations, Communications of the Kharkov Mathematical Society. The second series 13 (1912) 200–224, in Russian.
  7. V. A. Dobrovolsky, J. Strelcyn, N. V. Lokot’, Mihail Nikolaevich Lagutinsky (1871– 1915), Istoriko-matematicheskie issledovaniya 6 (2001) 111–127, in Russian.
  8. C. Christopher, J. Llibre, J. Vito´rio Pereira, Multiplicity of Invariant Algebraic Curves in Polynomial Vector Fields, Pacific Journal of Mathematics 229 (1) (2007) 63–117. doi: 10.2140/pjm.2007.229.63.
  9. M. D. Malykh, On M. N. Lagutinsky’s Method for Computation of Rational Integrals of Ordinary Differential Equations Systems, Vestnik natsional’nogo issledovatel’skogo yadernogo universiteta “MIFI” 5 (4) (2016) 327–336, in Russian. doi: 10.1134/S2304487X16030068.
  10. M. D. Malykh, On application of m. n. lagutinski method to differential equations in symbolic form. part 1, RUDN Journal of Mathematics, Information Sciences and Physics 25 (2) (2017) 103–112, in Russian. doi: 10.22363/2312-9735-2017-25-2-103-112.
  11. A. F. Filippov, Text-Book on Differential Equations, R&C Dynamics, Izhevsk, 2000, in Russian.



Abstract - 356

PDF (Russian) - 188




Copyright (c) 2018 Malykh M.D., Yu Y.

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies