Lie algebras associated with scalar second‐order ordinary differential equations

1989 ◽  
Vol 30 (12) ◽  
pp. 2770-2777 ◽  
Author(s):  
F. M. Mahomed ◽  
P. G. L. Leach
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Second Order

scholarly journals Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160461 ◽  
Author(s):  
A. A. Gainetdinova ◽  
R. K. Gazizov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Transformation Group ◽  
Second Order ◽  
Differential Invariants ◽  
Invariant Representation ◽  
Invariant Differentiation

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

Symmetry Lie algebras and properties of linear ordinary differential equations with maximal dimension

2015 ◽  
Vol 38 ◽  
pp. 1560074
Author(s):  
Mensah Folly-Gbetoula ◽  
A. H. Kara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Second Order ◽  
Maximal Dimension ◽  
Linear Ordinary Differential Equations ◽  
First Order ◽  
Iterative Equations ◽  
Logarithmic Type

Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the first-order source equation are obtained. Based on certain properties of iterative equations, finding the solutions is reduced to finding solutions of the second-order source equation. We have therefore found classes of solutions to the source equations by letting the parameters of the source equation be functions of a specific type such as monomials, functions of exponential and logarithmic type.

scholarly journals Linearization from Complex Lie Point Transformations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Sajid Ali ◽  
M. Safdar ◽  
Asghar Qadir
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Complex Structure ◽  
Second Order ◽  
Geometrical Construction ◽  
Maximum Dimension ◽  
Point Transformations

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimensiond, withd≤4. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue inR3of the linearizability criteria inR2.

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