The Calculation and Use of Generalized Symmetries for Second-Order Ordinary Differential Equations

  • C. MurielEmail author
  • J. L. RomeroEmail author
  • A. RuizEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)


New relationships between generalized symmetries, commuting symmetries, and generalized \(\mathscr {C}^\infty \)–symmetries for second-order ordinary differential equations are established. The sets of solutions of the respective determining equations are interrelated, which provides new strategies for solving them. Particular solutions of these determining equations can be appropriately combined in order to provide first integrals and Jacobi last multipliers for the equation.


Differential equation Generalized symmetry First integral Jacobi last multiplier 



The authors acknowledge the financial support from the University of Cádiz by means of the project PR2017-090 and from the Junta de Andalucía research group FQM-377.

A. Ruiz acknowledges the financial support from the Ministerio de Educación, Cultura y Deporte of Spain by means of a FPU grant (FPU15/02872).


  1. 1.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)CrossRefGoogle Scholar
  2. 2.
    Duarte, L.G.S., Moreira, I.C., Santos, F.C.: Linearization under non-point transformations. J. Phys. A Math. Gen. 27, L739–L743 (1994)CrossRefGoogle Scholar
  3. 3.
    Guha, P., Choudhury, A., Khanra, B.: \(\lambda -\)Symmetries, isochronicity and integrating factors of nonlinear ordinary differential equations. J. Eng. Math. 82(1), 85–99 (2013). Scholar
  4. 4.
    Ibragimov, N.H.: A Practical Course in Differential Equations and MathematicalModelling: Classical and New Methods, Nonlinear Mathematical Models Symmetry and Invariance Principles. World Scientific, Beijing (2010)Google Scholar
  5. 5.
    Ince, E.: Ordinary Differential Equations. Dover, New York (1956)Google Scholar
  6. 6.
    Muriel, C., Romero, J.: The \(\lambda \)-symmetry reduction method and Jacobi last multipliers. Commun. Nonlinear Sci. Numer. Simul. 19(4), 807–820 (2014)Google Scholar
  7. 7.
    Muriel, C., Romero, J.L.: New methods of reduction for ordinary differential equations. IMA J. Appl. Math. 66(2), 111–125 (2001). Scholar
  8. 8.
    Muriel, C., Romero, J.L.: Second-order ordinary differential equations and first integrals of the form \(A(t, x)\dot{x}+B(t, x)\). J. Nonlinear Math. Phys. 16(1), 209–222 (2009). Scholar
  9. 9.
    Muriel, C., Romero, J.L.: Nonlocal transformations and linearization of second-order ordinary differential equations. J. Phys. A Math. Theor. 43(43), 434,025 (13 pp) (2010). Scholar
  10. 10.
    Muriel, C., Romero, J.L.: \(\lambda \)-symmetries of some chains of ordinary differential equations. Nonlinear Anal. Real World Appl. 16, 191–201 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Muriel, C., Romero, J.L., Ruiz, A.: \(\lambda \)-symmetries and integrability by quadratures. IMA J. Appl. Math. 82(5), 1061–1087 (2017). Scholar
  12. 12.
    Nucci, M.: Jacobi last multiplier and Lie symmetries: a novel application of an old relationship. J. Nonlinear Math. Phys. 12(2), 284–304 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107. Springer, New York (1986). Scholar
  14. 14.
    Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic, New York (1982). [Harcourt Brace Jovanovich Publishers]. Translated from the Russian by Y. Chapovsky, Translation edited by William F. AmesGoogle Scholar
  15. 15.
    Stephani, H.: Differential Equations: Their Solution Using Symmetries. Cambridge University, Cambridge (1989)Google Scholar
  16. 16.
    Whittaker, E., McCrae, W.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge Mathematical Library. Cambridge University, Cambridge (1988)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departmento de MatemáticasUniversidad de CádizPuerto RealSpain

Personalised recommendations