
SIGMA 9 (2013), 041, 10 pages arXiv:1302.3242
http://dx.doi.org/10.3842/SIGMA.2013.041
On the Linearization of SecondOrder Ordinary Differential Equations to the Laguerre Form via Generalized Sundman Transformations
M. Tahir Mustafa, Ahmad Y. AlDweik and Raed A. Mara'beh
Department of Mathematics & Statistics, King University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Received February 16, 2013, in final form May 25, 2013; Published online May 31, 2013
Abstract
The linearization problem for nonlinear secondorder
ODEs to the Laguerre form by means of generalized Sundman
transformations (Stransformations) is considered, which has been
investigated by Duarte et al. earlier. A characterization of these
Slinearizable equations in terms of first integral and procedure
for construction of linearizing Stransformations has been given
recently by Muriel and Romero. Here we give a new characterization of
Slinearizable equations in terms of the coefficients of ODE and
one auxiliary function. This new criterion is used to obtain the
general solutions for the first integral explicitly, providing a
direct alternative procedure for constructing the first integrals
and Sundman transformations. The effectiveness of this approach is
demonstrated by applying it to find the general solution for
geodesics on surfaces of revolution of constant curvature in a
unified manner.
Key words:
linearization problem; generalized Sundman transformations; first integrals; nonlinear secondorder ODEs.
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References
 Bluman G.W., Anco S.C., Symmetry and integration methods for differential
equations, Applied Mathematical Sciences, Vol. 154, SpringerVerlag,
New York, 2002.
 Chandrasekar V.K., Senthilvelan M., Lakshmanan M., A unification in the theory
of linearization of secondorder nonlinear ordinary differential equations,
J. Phys. A: Math. Gen. 39 (2006), L69L76,
nlin.SI/0510045.
 Chandrasekar V.K., Senthilvelan M., Lakshmanan M., On the complete
integrability and linearization of certain secondorder nonlinear ordinary
differential equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng.
Sci. 461 (2005), 24512476, nlin.SI/0408053.
 Duarte L.G.S., Moreira I.C., Santos F.C., Linearization under nonpoint
transformations, J. Phys. A: Math. Gen. 27 (1994),
L739L743.
 Euler N., Transformation properties of x''+f_{1}(t)x'+f_{2}(t)x+f_{3}(t)x^{n}=0, J. Nonlinear Math. Phys. 4 (1997),
310337.
 Euler N., Euler M., Sundman symmetries of nonlinear secondorder and
thirdorder ordinary differential equations, J. Nonlinear Math.
Phys. 11 (2004), 399421.
 Euler N., Wolf T., Leach P.G.L., Euler M., Linearisable thirdorder ordinary
differential equations and generalised Sundman transformations: the case
X'''=0, Acta Appl. Math. 76 (2003), 89115,
nlin.SI/0203028.
 Ibragimov N.H., Practical course in differential equations and mathematical
modelling, ALGA, Karlskrona, 2006.
 Ibragimov N.H., Magri F., Geometric proof of Lie's linearization theorem,
Nonlinear Dynam. 36 (2004), 4146.
 Mahomed F.M., Symmetry group classification of ordinary differential equations:
survey of some results, Math. Methods Appl. Sci. 30 (2007),
19952012.
 Meleshko S.V., On linearization of thirdorder ordinary differential equations,
J. Phys. A: Math. Gen. 39 (2006), 1513515145.
 Mimura F., Nôno T., A new conservation law for a system of secondorder
differential equations, Bull. Kyushu Inst. Tech. Math. Natur. Sci.
(1994), no. 41, 110.
 Muriel C., Romero J.L., Nonlocal transformations and linearization of
secondorder ordinary differential equations, J. Phys. A: Math.
Theor. 43 (2010), 434025, 13 pages.
 Muriel C., Romero J.L., Secondorder ordinary differential equations and first
integrals of the form A(t,x)x'+B(t,x), J. Nonlinear Math.
Phys. 16 (2009), suppl. 1, 209222.
 Muriel C., Romero J.L., Secondorder ordinary differential equations with first
integrals of the form C(t)+1/(A(t,x)x'+B(t,x)), J. Nonlinear
Math. Phys. 18 (2011), suppl. 1, 237250.
 Nakpim W., Meleshko S.V., Linearization of secondorder ordinary differential
equations by generalized Sundman transformations, SIGMA 6
(2010), 051, 11 pages, arXiv:1006.2891.
 Pressley A., Elementary differential geometry, 2nd ed., Springer Undergraduate
Mathematics Series, SpringerVerlag, London, 2010.
 Stephani H., Differential equations. Their solution using symmetries, Cambridge
University Press, Cambridge, 1989.

