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On Differential Invariants and Classification of Ordinary Differential Equations of the Form y'' = A(x, y)y' + B(x, y)

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Abstract

The class of second-order ordinary differential equations y'' = A(x, y)y' + B(x, y) is studied by methods of the geometry of jet spaces and the geometric theory of differential equations. The symmetry group of this class of equations is calculated, and the field of differential invariants of its action on equations is described. These results are used to state and prove a criterion for the local equivalence of two nondegenerate ordinary differential equations of the form y'' = A(x, y)y' + B(x, y), inwhich the coefficients A and B are rational in x and y.

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References

  1. B. S. Kruglikov, “Point classification of second order ODEs: Tresse classification revisited and beyond,” in Differential Equations: Geometry, Symmetries and Integrability (Springer, Berlin, 2009).

    Chapter  Google Scholar 

  2. A. Tresse, Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre y'' = ω(x, y, y') (Leipzig, 1896).

    MATH  Google Scholar 

  3. R. A. Sharipov, Effective Procedure of Point Classification for the Equation y'' = P + 3Qy' + 3Ry'2 + Sy'3, arXiv: 9802027 (1998).

    Google Scholar 

  4. S. Lie, “Klassification und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten. III,” Lie Arch. 8, 371–458 (1883).

    MATH  Google Scholar 

  5. R. Liouville, “Sur les invariants de certaines equations differentielles et sur leurs applications,” J. de l’Ećole Polytech. 59, 7–76 (1889).

    MATH  Google Scholar 

  6. S. Yu. Slavyanov, “Polynomial degree reduction of a Fuchsian 2 × 2 system,” Teoret. Mat. Fiz. 182 (2), 223–230 (2015) [Theoret. and Math. Phys. 182 (2), 182–188 (2015)].

    Article  MathSciNet  Google Scholar 

  7. B. Dubrov, “Contact trivialization of ordinary differential equations,” in Differential Geometry and Its Applications (Silesian Univ. Opava, Opava, 2001), pp. 73–84.

    Google Scholar 

  8. B. Dubrov, “On a class of contact invariants of systems of ordinary differential equations,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 1, 76–77 (2006) [RussianMath. (Iz. VUZ) 50 (1), 73–74 (2006)].

    MathSciNet  Google Scholar 

  9. A. Kushner, Classifications of Monge–Ampère Equations, Doctoral Dissertation in Physics and Mathematics (Kazan State Univ., Kazan, 2011).

    Google Scholar 

  10. A. Kushner, V. Lychagin, and V. Rubtsov, Contact Geometry and Non-Linear Differential Equations (Cambridge Univ. Press, Cambridge, 2007).

    MATH  Google Scholar 

  11. D. V. Alekseevskii, A. M. Vinogradov, and V. V. Lychagin, Basic Ideas and Concepts of Differential Geometry, in Current Problems inMathematics: Fundamental Directions, Vol. 28, Itogi Nauki i Tekhniki (VINITI, Moscow, 1988), pp. 5–289 [in Russian].

    Google Scholar 

  12. A. M. Vinogradov, I. S. Krasil’shchik, and V. V. Lychagin, Introduction to the Geometry of Nonlinear Differential Equations (Nauka, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  13. P. V. Bibikov, “On Lie’s problem and differential invariants of ODEs y'' = F(x, y),” Funktsional. Anal. Prilozhen. 51 (4), 16–25 (2017) [Functional Anal. Appl. 51 (4), 255–262 (2017)].

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Bibikov and A. Malakhov, “On Lie problem and differential invariants for the subgroup of the plane Cremona group,” J. Geom. Phys. 121, 72–82 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Bibikov, “Generalized Lie problem and differential invariants for the third order ODEs,” Lobachevskii J. Math. 38 (4), 622–629 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations (Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2000) [in Russian].

    Google Scholar 

  17. E. Cartan, “Sur les variétés àconnexion projective,” Bull. Soc.Math. France 52, 205–241 (1924).

    Article  MathSciNet  Google Scholar 

  18. V. A. Yumaguzhin, Differential Invariants of 2-Order ODEs, I, arXiv: 0804.0674.

  19. B. Kriglikov and V. Lychagin, “Global Lie–Tresse theorem,” Selecta Math. (N. S.) 22 (3), 1357–1411 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  20. P. V. Bibikov, “Point equivalence of functions on the 1-jet space J1R,” Funktsional. Anal. Prilozhen. 48 (4), 19–25 (2014) [Functional Anal. Appl. 48 (4), 250–255 (2014)].

    Article  MathSciNet  Google Scholar 

  21. É. Vinberg and V. Popov, The Theory of Invariants (VINITI, Moscow, 1989) [in Russian].

    Google Scholar 

  22. M. Rosenlicht, “A remark on quotient spaces,” An. Acad. Brasil. Ci. 35, 487–489 (1963).

    MathSciNet  MATH  Google Scholar 

  23. Computer Algebra: Symbolic and Algebraic Computation, 2nd ed., Ed. by B. Buchberger, G. E. Collins, and R. Loos in cooperation with R. Albrecht (Springer-Verlag, Vienna, 1983;Mir,Moscow, 1986).

  24. V. V. Lychagin, “Feedback differential invariants,” Acta Appl.Math. 109 (1), 211–222 (2010).

    Article  MathSciNet  MATH  Google Scholar 

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  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    P. V. Bibikov

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Original Russian Text © P.V. Bibikov, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 2, pp. 163–173.

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Bibikov, P.V. On Differential Invariants and Classification of Ordinary Differential Equations of the Form y'' = A(x, y)y' + B(x, y). Math Notes 104, 167–175 (2018). https://doi.org/10.1134/S0001434618070180

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