Classification of Scalar Fourth Order Ordinary Differential Equations Linearizable via Generalized Lie–Bäcklund Transformations

  • Hina M. DuttEmail author
  • Asghar Qadir
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)


Lie had shown that there is a unique class of scalar second order ordinary differential equations (ODEs) that can be converted to linear form by point transformations. Mahomed and Leach had shown that for higher order (than 2) scalar ODEs there are always three classes. Separately, Chern had linearized two classes of third order ODEs by using contact transformations. We provided an (inclusive) classification for third order ODEs by using a generalization of contact transformations. Here we extend that work to the fourth order using a generalization of the Lie–Backlund transformation and demonstrate that there are (at least) four classes of fourth order linearizable ODEs.


Generalized Lie–Bäcklund transformations Linearization System of two second order ODEs 



We are grateful to the Higher Education Commission (HEC) of Pakistan for support under their project no. 3054.


  1. 1.
    Chern, S.S.: Sur la géométrie d’une équation différentielle du troisème orde. C.R. Acad. Sci. Paris 204, 1227–1229 (1937)Google Scholar
  2. 2.
    Chern, S.S.: The geometry of the differential equation \( y^{^{\prime \prime \prime }}=F(x, y, y^{^{\prime }}, y^{^{\prime \prime }})\). Sci. Rep. Nat. Tsing Hua Univ. 4, 97–111 (1940)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dutt, H.M., Qadir, A.: Classification of scalar third order ordinary diferential equations linearizable via generalized contact transformations. Quaest. Math. 41(1), 15–26 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Grebot, G.: The linearization of third order ODEs, Preprint (1996)Google Scholar
  5. 5.
    Grebot, G.: The characterization of third order ordinary differential equations admitting a transitive fibre preserving point symmetry group. J. Math. Anal. Appl. 206, 364–388 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ibragimov, N.H., Meleshko, S.V.: Linearization of third order ordinary differential equations by point transformations. Arch. ALGA 1, 71–93 (2004)Google Scholar
  7. 7.
    Ibragimov, N.H., Meleshko, S.V.: Linearization of third order ordinary differential equations by point and contact transformations. J. Math. Anal. Appl. 308, 266–289 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ibragimov, N.H., Meleshko, S.V., Suksern, S.: Linearization of fourth order ordinary differential equations by point transformations. J. Phys. Math. Theor. 41(235206), 1–19 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Laguerre, E.: Sur les équations différentielles linéaires du troisième ordre. Comptes Rendus 88, 116–119 (1879)zbMATHGoogle Scholar
  10. 10.
    Laguerre, E.: Sur quelques invariants des èquations diffèrentielles. Comptes Rendus 88, 224–227 (1879)zbMATHGoogle Scholar
  11. 11.
    Lie, S.: Klassifikation und integration von gewöhnlichen differentialgleichungen zwischen x, y, die eine gruppe von transformationen gestatten I, II, II and IV. Archiv for Mathematik, and 9 187, 249, 371, 431 (1883). [Gesammelte Abhandlungen, V, 240, 282, 362 and 432]Google Scholar
  12. 12.
    Lie, S.: Theorie der Transformationsgruppen I, II and III. Teubner, Leipzig (1888). [Reprinted by Chelsea Publishing Company, New York (1970)]Google Scholar
  13. 13.
    Lie, S.: Differential Equations. Chelsea, New York (1967)Google Scholar
  14. 14.
    Lie, S.: Lectures on differential equations with known infinitesimal transformations. Teubner, Leipzig. [In German, Lie’s lectures by G. Sheffers (1891)]Google Scholar
  15. 15.
    Mahomed, F.M.: Symmetry group classification of ordinary differential equations: survey of some results. Math. Methods Appl. Sci. 30, 1995–2002 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mahomed, F.M., Leach, P.G.L.: Lie algebras associated with second order ordinary differential equations. J. Math. Phys. 30, 2770–2777 (1989)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mahomed, F.M., Leach, P.G.L.: Symmetry Lie algebra of \(n\)th order ordinary differential equations. J. Math. Anal. Appl. 151, 80–107 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Neut, S., Petitot, M.: La géométrie de l’équation \(y^{\prime \prime \prime }=f(x, y, y^{\prime }, y^{\prime \prime })\). C.R. Acad. Sci. Paris I 335, 515–518 (2002)CrossRefGoogle Scholar
  19. 19.
    Qadir, A.: Linearization of ordinary differential equations by using geometry. SIGMA 3, 1–7 (2007)Google Scholar
  20. 20.
    Suksern, S., Ibragimov, N.H., Meleshko, S.V.: Criteria for the fourth order ordinary differential equations to be linearizable by contact transformations. Commun. Nonlinear Sci. Numer. Simul. 14, 2619–2628 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tressé, A.: Sur les Invariants differentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wafo, Soh C., Mahomed, F.M.: Linearization criteria for a system of second order ordinary differential equations. Int. J. Non-Linear Mech. 36, 671–677 (2001)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Basic Sciences, School of Electrical Engineering and Computer ScienceNational University of Sciences and TechnologyIslamabadPakistan
  2. 2.Physics Department School of Natural SciencesNational University of Sciences and TechnologyIslamabadPakistan

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