A new way to use nonlocal symmetries to determine first integrals of second-order nonlinear ordinary differential equations

Finding first integrals of second-order nonlinear ordinary differential equations (nonlinear 2ODEs) is a very difficult task. In very complicated cases, where we cannot find Darboux polynomials (to construct an integrating factor) or a Lie symmetry (that allows us to simplify the equations), we sometimes can solve the problem by using a nonlocal symmetry. In [1], [2], [3] we developed (and improved) a method (S-function method) that is successful in finding nonlocal Lie symmetries to a large class of nonlinear rational 2ODEs. However, even with the nonlocal symmetry, we still need to solve a 1ODE (which can be very difficult to solve) to find the first integral. In this work we present a novel way of using the nonlocal symmetry to compute the first integral with a very efficient linear procedure.

New version program summary

Program Title: InSyDE – Invariants and Symmetries of (rational second order ordinary) Differential Equations.

CPC Library link to program files: https://doi.org/10.17632/4ytft6zgk7.3

Licensing provisions: CC by NC 3.0

Programming language: Maple

Supplementary material: Theoretical results and revision of the S-function method.

Journal reference of previous version: Comput. Phys. Comm. Volume 234, January 2019, Pages 302-314 - https://doi.org/10.1016/j.cpc.2018.05.009

Does the new version supersede the previous version?: Yes.

Nature of problem: Determining first integrals of rational second order ordinary differential equations.

Solution method: The method is explained in the Summary of revisions and Supplementary material.

Reasons for the new version: The InSyDE package after determining the S-function still needs to solve a first-order ordinary differential equation (1ODE) associated with the nonlocal symmetry (the so-called associated 1ODE – see [2]). The problem is that, for very complicated 1ODEs, this may not be practically feasible. We have developed an new and more efficient method that uses the nonlocal symmetry to (for a large class of 1ODEs) determine the first integral in a linear way.

Summary of revisions: In order to implement the new method just mentioned above we have made modifications to the command (Sfunction) and introduced a new one: command (Darlin).