Lobachevskii Journal of Mathematics

, Volume 34, Issue 3, pp 264–271 | Cite as

On the contact equivalence problem of second order ODEs which are quadratic with respect to the second order derivative



In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form y2 + A(x, y, y′)y″ + B(x, y, y′) = 0 to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional distributions.

Keywords and phrases

Contact transformations point transformations differential invariants ODEs 


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  1. 1.
    V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, Berlin-Heidelberg, 2008).Google Scholar
  2. 2.
    N. Ibragimov, A practical course in differential equations and mathematical modelling. Classical and new methods, nonlinear mathematical models, symmetry and invariance principles (Higher Education Press, Beijing and World Scientific Singapore, 2009).CrossRefGoogle Scholar
  3. 3.
    N. Ibragimov and F. Magri, Nonlinear Dynamics 36, 41 (2004).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    B. Kruglikov, Point classification of 2nd order ODEs: Tresse classification revisited and beyond in Differential equations: Geometry, Symmetries and Integrability: The Abel Symposium 2008, Abel Symposia 5 (Berlin, Springer, 2009).Google Scholar
  5. 5.
    S. Lie, Gesam. Abh. Bd. 5, paper XIV, 362 (1924).Google Scholar
  6. 6.
    R. Liouville, Journal de l’École Polytechnique 59, 7 (1889).Google Scholar
  7. 7.
    O. I. Morozov, Nauch. Vestn. MGTU GA 157, 90 (2006).Google Scholar
  8. 8.
    A. Tresse, ActaMath. 18, 1 (1894).MathSciNetMATHGoogle Scholar
  9. 9.
    A. Tresse, Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre y″ = ω(x, y, y′) (Leipzig, 1896).MATHGoogle Scholar
  10. 10.
    V. Yumaguzhin, Acta Applicandae Mathematicae 83(1–2), 133 (2004).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    V. Yumaguzhin, Acta Applicandae Mathematicae 109, 283 (2010).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia

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