Integrating factor, first integral and λ-symmetry for vector ODEs of second orders

Subject Areas : Applied Mathematics

Khodayar Goodarzi ^{1 *}

1 - Department of Mathematics, Brujerd Branch, Islamic Azad University, Broujerd, Iran

Received: 2022-12-29 Accepted : 2024-04-28 Published : 2024-09-22

Keywords: Symmetry, λ-symmetry, λ-prolongation, integrating factor, first integral.,

Abstract :

The goal of this paper is to calculate an integrating factor, λ-symmetry and a
first integral for ODEs of second order ¨u = F(x, u, u˙) and for a vector ODEs
of second order ¨rˆ = ˆf, by the λ-symmetry method.

 

References:

1] B. Abraham-Shrdauner B, Hidden symmetries and non-local group
generators for ordinary differential equation, IMA J. Appl. Math. 56(1996)
235–252.
[2] V. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Extended
Prelle–Singer method and integrability/solvability of a class of nonlinear
nth-order ordinary differential equations, J. Nonl. Math.Phys. 12(2005)
84–201.
[3] V. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Unusual Lienardtype nonlinear oscillator, Physical Review E, 72(2005) Article ID 066203,
8 pages.
[4] G. Gaeta and P. Morando, On the geometry of lambda-symmetries and
PDEs reduction, J. Phys. A 37(2004) 6955–6975.
[5] C. Muriel and J.L. Romero, New methods of reduction for ordinary
differential equation, IMA J. Appl. Math. 66 (2001) 111–125.
[6] C. Muriel and J.L. Romero, C∞-symmetries and reduction of equation
without Lie point symmetries, J. Lie Theory 13 (2003) 167–188.
7] Muriel C and Romero J, 2008 λ-symmetries and integrating factors, J.
Non-linear Math. Phys.15(2008)300–309.
[8] C. Muriel and J.L. Romero, 2009 First integrals, integrating factors and
λ-symmetries of second-order differential equations, J. Phys. A:Math.
Theor.43(2009) 365207.
[9] C. Muriel and J.L. Romero, 2009 Second-order ordinary differential
equations and first integral of the form A(t, x) ˙x + B(t, x) J. Nonlinear
Math. Phys.16(2009) 209–222.
[10] C. Muriel and J.L. Romero, A λ-symmetry-based method for the
linearization and determination of first-integrals of a family of second-order
differential equations, J. Phys. A:Math. Theor.44(2011) 201–245.
[11] P.J. Olver, Applications of Lie Groups to Differential Equations, (New
York, 1986).