Complete group classification of systems of two linear second‐order ordinary differential equations: the algebraic approach

Abstract

We give a complete group classification of the general case of linear systems of two second-order ordinary differential equations. The algebraic approach is used to solve the group classification problem for this class of equations. This completes the results in the literature on the group classification of two linear second-order ordinary differential equations including recent results which give a complete group classification treatment of such systems. We show that using the algebraic approach leads to the study of a variety of cases in addition to those already obtained in the literature. We illustrate that this approach can be used as a useful tool in the group classification of this class of equations. A discussion of the subsequent cases and results is given.

On group classification of normal systems of linear second-order ordinary differential equations

“…This result has been stated as a Theorem in Section 5 of the paper. The results were discussed and in two or more of the cases which had the admitted generators of the form X A the study was similar to that done in the earlier studies in [1,2,3,4]. A detailed discussion on systems admitting generators of the form X A has been given in Section 6 of the paper.…”

“…Multiplying from the left hand side by e xA 3 and by e −xA 3 from the right hand side, equations (19) can be rewritten as…”

“…This has been part of the reason for focusing on such systems in the current study. We study the general group classification of systems of linear secondorder ordinary differential equations motivated by recent results obtained in [1,2,3,4]. Linear equations play an important role in many applications where they occur in a disguised form.…”

“…Here we use the algebraic algorithm in the group classification approach. This approach was used in the earlier works [3,5,6,7] and references therein. The algebraic algorithm assists in simplifying the study of the group classification.…”

Canonical Forms and Their Integrability for Systems of Three 2nd-Order ODEs

“…After Lie, several researchers have been involved in analysis of different aspects of this work; see [1,6,[8][9][10][11][12]. Although the algebraic approach is more involved, it generates new cases and more insight to mathematical properties whose base is algebra (see [8,9,13,14]). Thus the classification of differential equations via algebraic approach is more efficient than other classical approaches.…”