Second-Order Ordinary Differential Equations and First Integrals of The Form A(t, x) ẋ + B(t, x)
- https://doi.org/10.1142/S1402925109000418How to use a DOI?
- Ordinary differential equations, first integrals, λ-symmetries, Sundman transformations
We characterize the equations in the class 𝒜 of the second-order ordinary differential equations ẍ = M(t, x, ẋ) which have first integrals of the form A(t, x)ẋ + 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)ẋ + B(t, x). The methods are applied to several relevant families of equations.
- © 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 -