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Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series

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The first integrals and rational solutions of some fourth-order differential equations

https://doi.org/10.29235/1561-2430-2020-56-3-318-327

Abstract

The object of this research is fourth-order differential equations. The aim of the research is to study the analytical properties of the solutions of these differential equations. The general form of the considered equations is indicated, and also the choice of the research object is justified. Herein we studied fourth-order differential equations for which sets of resonances with all positive nontrivial resonances are absent. Besides, three of these equations satisfy the conditions of absence in the solutions of moving multivalued singular points. The solutions of the next three equations have movable special points of multivalued character. Moreover, we also investigated the analytical properties of one more fourth-order differential equation of another general form for which it is also possible to construct a two-parameter rational solution as there is a nontrivial negative resonance in the related set of resonances. The first integrals of the equations under study are found and their rational solutions are constructed from negative non-trivial resonances. The resonance method was used in this study. The obtained results can be used in the analytical theory of differential equations.

About the Authors

E. R. Babich
Yanka Kupala State University of Grodno
Belarus

Elena R. Babich – Postgraduate Student of the Department of Mathematical Analysis, Differential Equations and Algebra

22, Ozheshko Str., 230023, Grodno



I. P. Martynov
Yanka Kupala State University of Grodno
Belarus

Ivan P. Martynov – Dr. Sc. (Physics and Mathematics), Professor, Professor of the Department of Mathematical Analysis, Differential Equations and Algebra

22, Ozheshko Str., 230023, Grodno



References

1. Ablowitz M. J., Ramani A., Segur H. A connection between nonlinear evolution equations and ordinary differential equation of P-type. Journal of Mathematical Physics, 1980, vol. 21, no. 4, pp. 715–721. https://doi.org/10.1063/1.524491

2. Clarkson P. A., Olver P. J. Symmetry and the Chazy Equation. Journal of Differential Equations, vol. 124, no. 1, pp. 225–246. https://doi.org/10.1006/jdeq.1996.0008

3. Zdunek A. G., Martynov I. P., Pronko V. A. On the rational solutions of differential equations. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta Imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal’naia Tekhnika i Kiravanne = Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol, 2000, no. 3, pp. 33–39 (in Russian).

4. Jrad F., Muğan U. Non-polynomial fourth order equations which pass the Painlevé test. Zeitschrift für Naturforschung A, 2005, vol. 60, no. 6, pp. 387–400. https://doi.org/10.1515/zna-2005-0601

5. Sobolevskii S. L. Movable Singular Points of Solutions of Ordinary Differential Equations. Minsk, 2008. 28 p. (in Russian).

6. Martynov I. P. On differential equations with fixed critical singular points. Differential Equations, 1973, vol. 9, no. 10, pp. 1780–1791 (in Russian).

7. Ablowitz M. J., Ramani A., Segur H. A connection between nonlinear evolution and ordinary differential equations of P-type. II. Journal of Mathematical Physics, 1980, vol. 21, no. 5, pp. 1006–1015. https://doi.org/10.1063/1.524548

8. Vankova T. N. Martynov I. P., Parmanchuk O. N., Pronko V. A. Some analytical properties of solutions of differential-algebraic equations. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta Imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal’naia Tekhnika i Kiravanne= Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol, 2008, no. 1, pp. 8–16 (in Russian).

9. Chzhan Bin’bin’, Martynov I. P. On rational solutions of a class of fourth-order non-polynomial differential equations. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta Imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal’naia Tekhnika i Kiravanne = Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol, 2018, vol. 8, no. 2, pp. 32–40 (in Russian).

10. Martynov I. P., Berezkina N. S., Pronko V. A. Analytical Theory of Nonlinear Equations and Systems. Grodno, GrGU Publ., 2009. 395 p. (in Russian).

11. Vankova T. N. Analytic Properties of Solutions of Some Classes of Differential Equations of the Third and Higher Order. Grodno, 2013. 105 p. (in Russian).

12. Martynov I. P. The equations of the third order with no moving critical features. Differential Equations, 1985, vol. 21, no. 6, pp. 937–946 (in Russian).

13. Kolesnikova N. S., Lukashevich N. A. On a class of third-order differential equations with fixed critical points. Differentsial’nye uravneniya = Differential Equations, 1972, vol. 8, no. 11, pp. 2082–2086 (in Russian).

14. Adjabi Y., Jrad F., Kessi A., Muğan U. Third order differential equations with fixed critical points. Applied Mathematics and Computation, 2009, vоl. 208, no. 1, pp. 238–248. https://doi.org/10.1016/j.amc.2008.11.044

15. Gantmacher F. R. Matrix Theory. Moscow, Nauka Publ., 1988. 548 p. (in Russian).


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ISSN 1561-2430 (Print)
ISSN 2524-2415 (Online)