Differential Invariants and First Integrals of the System of Two Linear Second-Order Ordinary Differential Equations
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
Full Text:
PDFReferences
[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)
Reminder
If you have already registered in Journal A and plan to submit article(s) to Journal B, please click the "CATEGORIES", or "JOURNALS A-Z" on the right side of the "HOME".
We only use the follwoing mailboxes to deal with issues about paper acceptance, payment and submission of electronic versions of our journals to databases:
pam@cscanada.org
pam@cscanada.net
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