Differential Invariants and First Integrals of the System of Two Linear Second-Order Ordinary Differential Equations

Yu. Yu. Bagderina

Abstract


In a recent paper the basis of algebraic invariants of the system of two linear second-order ordinary di_erential equations has been found. Now we obtain the di_erential invariants for this family of equations, which depend on the _rst-order derivatives. It is shown that the _rst integrals of such systems can be sought as the functions of the algebraic and di_erential invariants of a given system. Di_erential invariants can be useful also in constructing the transformation connecting two equivalent systems when their algebraic invariants are constant.


Keywords


Dierential invariant; Equivalence; Linear equations; First integral; Invariant

Full Text:

PDF

References


[1] Wafo Soh, C., & Mahomed, F. M. (2000). Symmetry breaking for a system of two linear second-order ordinary

differential equations. Nonlin. Dynamics, 22, 121-133.

[2] Wafo Soh, C. (2010). Symmetry breaking of systems of linear second-order ordinary differential equations with

constant coeffcients. Comm. Nonlin. Sci. Numer. Simul., 15, 139-143.

[3] Meleshko, S. V. (2011). Comment on “Symmetry breaking of systems of linearsecond-order ordinary differential

equations with constant coe_cients”. Comm. Nonlin. Sci. Numer. Simul., 16, 3447-3450.

[4] Moyo, S., Meleshko, S. V., & Oguis, G. F. (2013). Complete group classi_cation of systems of two linear second-

order ordinary di_erential equations. Comm. Nonlin. Sci. Numer. Simul., 18, 2972-2983.

[5] Meleshko, S. V., Moyo, S., & Oguis, G. F. (2014). On the group classiffcation of systems of two linear second-

order ordinary differential equations with constant coeffcients. J. Math. Analysis Appl., 410, 341-347.

[6] Campoamor-Stursberg, R. (2011). Systems of second-order linear ODE's with constant coe_cients and their

symmetries. Comm. Nonlin. Sci. Numer. Simul., 16, 3015-3023.

[7] Campoamor-Stursberg, R. (2012). Systems of second-order linear ODE's with constant coe_cients and their

symmetries. ii. Comm. Nonlin. Sci. Numer. Simul., 17, 1178-1193.

[8] Wilczynski, E. J. (1906). Projective di_erential geometry of curves and ruled surfaces. Leipzig: Teubner.

[9] Bagderina, Yu. Yu. (2011). Equivalence of linear systems of two second-order ordinary differential equations.

Progress in Applied Mathematics, 1, 106-121.

[10] Gonzalez-Lopez, A. (1988). Symmetries of linear systems of second-order ordinary

differential equations. J. Math. Phys., 29, 1097-1105.

[11] Gorringe, V. M., & Leach, P. G. L. (1988). Lie point symmetries for systems ofsecond order linear ordinary

differential equations. Quaestiones Mathematicae, 11, 95-117.

[12] Boyko, V. M., Popovych, R. O., & Shapoval, N. M. (2013). Lie symmetries of systems of second-order linear

ordinary differential equations with constant coeffcients. J. Math. Analysis Appl., 397, 434-440.

[13] Bagderina, Yu. Yu. (2014). Symmetries and invariants of the systems of two linearsecond-order ordinary

differential equations. Comm. Nonlin. Sci. Numer. Simul., 19, 3513-3522.

[14] Wafo Soh, C., & Mahomed, F. M. (2001). Linearization criteria for a system of second-order ordinary di_erential

equations. Int. J. Non-Linear Mech., 36, 671-677.

[15] Bagderina, Yu. Yu. (2010). Linearization criteria for a system of two second-order ordinary di_erential equations.

J. Phys. A: Math. Theor., 43, 465201.

[16] Bagderina, Yu. Yu. (2013). Invariants of a family of scalar second-order ordinary differential equations. J. Phys.

A: Math. Theor, 46, 295201.

[17] Ovsiannikov, L. V. (1982). Group analysis of di_erential equations. New York:Academic Press.

[18] Ibragimov, N. H. (2004). Equivalence groups and invariants of linear and nonlinear equations. Arch ALGA, 1,

9-69.

[19] Barinov, V. A., & Butakova, N. N. (2004). Wave propagation over the free surface of a two-phase medium with a

nonuniform concentration of the disperse phase. J. Appl. Mech. Tech. Phys., 45, 477-485.




DOI: http://dx.doi.org/10.3968/4825

DOI (PDF): http://dx.doi.org/10.3968/pdf_8

Refbacks

  • There are currently no refbacks.


Copyright (c)



Share us to:   

Reminder

We are currently accepting submissions via email only.

The registration and online submission functions have been disabled.

Please send your manuscripts to pam@cscanada.net，or   pam@cscanada.org  for consideration.

We look forward to receiving your work.

 

 

 Articles published in Progress in Applied Mathematics are licensed under Creative Commons Attribution 4.0 (CC-BY).

 ROGRESS IN APPLIED MATHEMATICS Editorial Office

Address: 1055 Rue Lucien-L'Allier, Unit #772, Montreal, QC H3G 3C4, Canada.

Telephone: 1-514-558 6138
Http://www.cscanada.net
Http://www.cscanada.org
E-mail:office@cscanada.net office@cscanada.org caooc@hotmail.com

Copyright © 2010 Canadian Research & Development Center of Sciences and Cultures