Advertisement

The Properties of Certain Linear and Nonlinear Differential Equations

Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)

Abstract

We consider linear differential equations of the second- and the third-order and nonlinear second-order differential equations related via the Schwarzian derivative. The main objective of the paper is to obtain relations between the solutions of the second- and the third-order linear differential equations and the solutions of the nonlinear differential equations of the second order. The method is based on the use of the Schwarzian derivative, which is defined as the ratio of two linearly independent solutions of the linear differential equations of the second and the third order. As a result, we obtain new relations between the solutions of these linear and nonlinear equations.

Notes

Acknowledgements

GF is grateful to the organizers of the conference Modern Mathematical Methods and High Performance Computing in Science and Technology—2018 for their invitation and the opportunity to present a recorded video lecture.

GF also acknowledges the support of the Alexander von Humboldt Foundation and the support of NCN OPUS 2017/25/B/BST1/00931.

References

  1. 1.
    Ahlfors, L. V.: Möbius transformations in several dimensions. Lecture notes at the University of Minnesota, Minneapolis (1981)zbMATHGoogle Scholar
  2. 2.
    Dobrovolsky, V.A.: Essays on the development of the analytic theory of differential equations. Vishcha Shkola, Kiev (1974) (in Russian)Google Scholar
  3. 3.
    Lukashevich, N. A.: On the third order linear equations. Differ. Equ., 35, 1384–1390 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Lukashevich, N. A., Martynov, I. P.: On the third order linear equations. Materials of the International Scientific Conference ”Differential Equations and Their Applications”, Grodno, Grodno State University, 78–85 (1998) (in Russian)Google Scholar
  5. 5.
    Lukashevich, N. A., Chichurin, A.V.: Differential equations of the first order. Belarusian State University, Minsk (1999) (in Russian)zbMATHGoogle Scholar
  6. 6.
    Ince, E. L.: Ordinary differential equations. Dover Publications, New York (1956)Google Scholar
  7. 7.
    Chichurin, A.V.: The Chazy equation and linear equations of the Fuchs class. Publ. RUDN, Moscow (2003) (in Russian)Google Scholar
  8. 8.
    Chichurin, A., Stepaniuk G.: The computer method of construction of the general solution of the linear differential equation of the third order. Studia i Materialy EWSIE, 8, 17–27 (2014)Google Scholar
  9. 9.
    Chichurin, A.V., Stepaniuk, G.P.: General solution to the fourth order linear differential equation with coefficients satisfying the system of three first order differential equations. Bulletin of Taras Shevchenko National University of Kyiv, Series: Physics & Mathematics, 31, No.1, 29–34 (2014)Google Scholar
  10. 10.
    Chichurin, A.V.: The computer method of construction of the general solution of the nonlinear differential equation of the fourth order. Studia i Materialy EWSIE, 7, 39–47 (2014)Google Scholar
  11. 11.
    Chichurin, A.: Computer construction of the general solution of the linear differential equation of the fifth order. Recent Developments in Mathematics and Informatics, Vol. I. Contemporary Mathematics and Computer Science, KUL, Lublin, 19-36 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Institute of Mathematics and Computer ScienceThe John Paul II Catholic University of LublinLublinPoland

Personalised recommendations