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On the integration of first order nonlinear differential equations and the conditions of Fuchs' theorem

Year 2022, Volume: 6 Issue: 3, 347 - 353, 30.09.2022
https://doi.org/10.31197/atnaa.1059343

Abstract

In this paper we give the general solutions of a class of first order nonlinear Fuchs ordinary differential equations. This leads us to show by an example that the necessary conditions of Fuchs' theorem are not sufficient.

References

  • [1] F.J. Bureau, differential equations with fixed critical points. Annali di Matematica pura ed applicata, 1964, LXIV, 229-364.
  • [2] R. Conte, The Painlevé approach to nonlinear ordinary differential equations, The Painlevé Property, Springer-Verlag, New York 1999, 77-180.
  • [3] R. Conte, Unification of PDE and ODE versions of Painlevé analysis into a single invariant version, Painlevé transcendents, their asymptotics and physical applications, eds. D. Levi and P. Winternitz, Plenum, New York, 1992, 125-144.
  • [4] R. Conte et al., Direct and inverse methods in nonlinear evolution equations. Springer-Verlag, Berlin 2003.
  • [5] L. Fuchs,über differentialgleichungen, deren integrale feste verzweigungspunkte besitzen. Koniglichen Preussischen Akademie der Wissenschaften, 1884, 32, 699-719.
  • [6] V.V. Golubev, Lectures on Analytic Theory of Differential Equations [in Russian], Gostekhteorizdat, Moscow, 1950. [7] C. Hermite, Course Lithographie de l'Ecole Polytechnique, Paris, 1873.
  • [8] E. Hille, Ordinary Differential Equations in the Complex Domai. Wiley-Interscience, New York, 1976.
  • [9] E. L. Ince, Ordinary Di?erential Equations. Dover, New york, 1964.
  • [10] N. Joshi and M.D. Kruskal, A local asymptotic method of seeing the natural barrier of the solutions of the Chazy equation, Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, ed. P.A. Clarkson, Plenum, New York, 1993, 331-340.
  • [11] A. Kessi, and K.M. Messaoud, First order equations without mobile critical points. Regular and chaotic dynamics, 2000, V. 6(1), 1-6.
  • [12] N.S. Kolesnikova, and N.A. Lukashevich, Sufficient conditions for the existence of solutions with stationary critical singular points for first order equations. Differencial'nye Uravnenija, 1972, V8(10), 1753-1760.
  • [13] K. M'hamed-Messaoud, A. Kessi, and T. Laadj, On Sufficient Conditions for the Existence of Solutions for First Order Equations and Fourth Degree with the Painlevé Property, Qualitative Theory of Dynamical Systems, 2016, V15(1), 81-93.
  • [14] K. M'hamed-Messaoud, T. Laadj and A. Kessi, First order fifth degree Fuchs differential equation with fixed critical points, Int. Journal of Dynamical Systems and Differential Equations, 2019 Vol.9 No.3, pp.286 - 297.
  • [15] P. Painlevé, Sur les lignes singulières des fonctions analytiques. Annales de la faculté de Toulouse, 1888.
  • [16] E. Picard, Sur l'emploi des approximations successives dans l'étude de certaines equations aux derivees partielles. Oeuvres de Emile Picard Paris 1979, Tome II, 377-383.
  • [17] A.D. Polyanin, V.F. Zaitsev, Handbook of exact solutions for ordinary differential equations, Boca Raton, Chapman &Hall/CRC, 2003.
  • [18] M.V. Soare, P.P. Teodorescu and I. Toma, Ordinary differential equations with applications to mechanics, Dordrecht, Springer, 2007.
Year 2022, Volume: 6 Issue: 3, 347 - 353, 30.09.2022
https://doi.org/10.31197/atnaa.1059343

Abstract

References

  • [1] F.J. Bureau, differential equations with fixed critical points. Annali di Matematica pura ed applicata, 1964, LXIV, 229-364.
  • [2] R. Conte, The Painlevé approach to nonlinear ordinary differential equations, The Painlevé Property, Springer-Verlag, New York 1999, 77-180.
  • [3] R. Conte, Unification of PDE and ODE versions of Painlevé analysis into a single invariant version, Painlevé transcendents, their asymptotics and physical applications, eds. D. Levi and P. Winternitz, Plenum, New York, 1992, 125-144.
  • [4] R. Conte et al., Direct and inverse methods in nonlinear evolution equations. Springer-Verlag, Berlin 2003.
  • [5] L. Fuchs,über differentialgleichungen, deren integrale feste verzweigungspunkte besitzen. Koniglichen Preussischen Akademie der Wissenschaften, 1884, 32, 699-719.
  • [6] V.V. Golubev, Lectures on Analytic Theory of Differential Equations [in Russian], Gostekhteorizdat, Moscow, 1950. [7] C. Hermite, Course Lithographie de l'Ecole Polytechnique, Paris, 1873.
  • [8] E. Hille, Ordinary Differential Equations in the Complex Domai. Wiley-Interscience, New York, 1976.
  • [9] E. L. Ince, Ordinary Di?erential Equations. Dover, New york, 1964.
  • [10] N. Joshi and M.D. Kruskal, A local asymptotic method of seeing the natural barrier of the solutions of the Chazy equation, Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, ed. P.A. Clarkson, Plenum, New York, 1993, 331-340.
  • [11] A. Kessi, and K.M. Messaoud, First order equations without mobile critical points. Regular and chaotic dynamics, 2000, V. 6(1), 1-6.
  • [12] N.S. Kolesnikova, and N.A. Lukashevich, Sufficient conditions for the existence of solutions with stationary critical singular points for first order equations. Differencial'nye Uravnenija, 1972, V8(10), 1753-1760.
  • [13] K. M'hamed-Messaoud, A. Kessi, and T. Laadj, On Sufficient Conditions for the Existence of Solutions for First Order Equations and Fourth Degree with the Painlevé Property, Qualitative Theory of Dynamical Systems, 2016, V15(1), 81-93.
  • [14] K. M'hamed-Messaoud, T. Laadj and A. Kessi, First order fifth degree Fuchs differential equation with fixed critical points, Int. Journal of Dynamical Systems and Differential Equations, 2019 Vol.9 No.3, pp.286 - 297.
  • [15] P. Painlevé, Sur les lignes singulières des fonctions analytiques. Annales de la faculté de Toulouse, 1888.
  • [16] E. Picard, Sur l'emploi des approximations successives dans l'étude de certaines equations aux derivees partielles. Oeuvres de Emile Picard Paris 1979, Tome II, 377-383.
  • [17] A.D. Polyanin, V.F. Zaitsev, Handbook of exact solutions for ordinary differential equations, Boca Raton, Chapman &Hall/CRC, 2003.
  • [18] M.V. Soare, P.P. Teodorescu and I. Toma, Ordinary differential equations with applications to mechanics, Dordrecht, Springer, 2007.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Arezki Kessi This is me 0000-0002-1381-7576

Toufik Laadj 0000-0002-1381-7576

Moussa Yahi This is me 0000-0002-1381-7576

Publication Date September 30, 2022
Published in Issue Year 2022 Volume: 6 Issue: 3

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