Journal of Nonlinear Mathematical Physics

Volume 16, Issue Supplement 1, March 2013, Pages 209 - 222

Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)

Authors
C. Muriel*, J. L. Romero
Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Cádiz, Spain
Corresponding Author
C. Muriel
Received 24 August 2009, Accepted 14 September 2009, Available Online 7 January 2021.
DOI
https://doi.org/10.1142/S1402925109000418How to use a DOI?
Keywords
Ordinary differential equations, first integrals, λ-symmetries, Sundman transformations
Abstract

We characterize the equations in the class 𝒜 of the second-order ordinary differential equations = M(t, x, ) which have first integrals of the form A(t, x) + B(t, x). We give an intrinsic characterization of the equations in 𝒜 and an algorithm to calculate explicitly such first integrals. Although 𝒜 includes equations that lack Lie point symmetries, the equations in 𝒜 do admit λ-symmetries of a certain form and can be characterized by the existence of such λ-symmetries. The equations in a well-defined subclass of 𝒜 can completely be integrated by using two independent first integrals of the form A(t, x) + B(t, x). The methods are applied to several relevant families of equations.

Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
16 - Supplement 1
Pages
209 - 222
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1142/S1402925109000418How to use a DOI?
Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - C. Muriel
AU  - J. L. Romero
PY  - 2021
DA  - 2021/01
TI  - Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)
JO  - Journal of Nonlinear Mathematical Physics
SP  - 209
EP  - 222
VL  - 16
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000418
DO  - https://doi.org/10.1142/S1402925109000418
ID  - Muriel2021
ER  -