Russian Mathematics

, Volume 62, Issue 11, pp 1–11 | Cite as

Classification of Certain Class of Ordinary Differential Equations of the First Order

  • P. V. BibikovEmail author
  • N. A. Safonkin


We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer.


ordinary differential equations differential invariant jet space algebraic geometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bibikov, P. “Differential Invariants and Contact Classification of Ordinary Differential Equations”, Lobachevskii J.Math. 36, No. 3, 245–249 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kruglikov, B. S. “Point Classification of Second Order ODEs: Tresse Classification Revisited and Beyond”, Proceedings of the Fifth Abel Symposium, Tromso, Norway, June 17–22 (Springer, Berlin, 2009), 199–221.Google Scholar
  3. 3.
    Bibikov, P. “On Lie Problem and Differential Invariants of the Equations y = F(x, y)”, Func. Analyzis and Appl. (in press).Google Scholar
  4. 4.
    Bibikov, P., Malakhov, A. “On Lie Problem and Differential Invariats for the Subgroup of the Plane Cremona Group”, J. Geom. and Phys. (in press).Google Scholar
  5. 5.
    Bibikov, P., Lychagin, V. “GL2(C)–Orbits of Binary Rational Forms”, Lobachevskii J. Math. 32, No. 1, 94–101 (2011).CrossRefzbMATHGoogle Scholar
  6. 6.
    Bibikov, P., Lychagin, V. “GL3(C)–Orbits of Rational Ternary Forms”, Dokl. Math. 84, No. 1, 482–484 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bibikov, P., Lychagin, V. “Classification of Linear Actions of Algebraic Groups on Spaces of Homogeneous Forms”, Dokl.Math. 85, No. 1, 102–109 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bibikov, P., Lychagin, V. “On Differential Invariants of Actions of Semisimple Lie Groups”, J. Geom. and Phys., issu. 85, 99–105 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bibikov, P., Lychagin, V. “Differential Contra Algebraic Invariants: Applications to Classical Algebraic Problems”, Lobachevskii J.Math. 37, No. 1, 36–49 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dolgachev, I., Iskovskikh, V. “Finite Subgroups of the Plane Cremona Group”, Algebra, Arithmetic, and Geom.: in honor of Yu. I.Manin. Vol. I, Progr.Math. 269 (Birkhuser, Boston, 2009), 443–548.CrossRefGoogle Scholar
  11. 11.
    Kruglikov, B., Lychagin, V. “Global Lie–Tresse Theorem”, Selecta Math. (N. S.. 22, No. 3, 1357–1411 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Alekseevskii, D. V., Vinogradov, A. M., Lychagin, V. V. “Basic Ideas and Notions From Differential Geometry”, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniy. 28, 1–299 (1988).[in Russian].zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Institute of Control Sciences of Russian Academy of SciencesMoscowRussia
  2. 2.National Research University “Higher School of Economics”MoscowRussia

Personalised recommendations