Reduction of Second-Order Ordinary Differential Equations into General Linear Equations via Point Transformations and its Application

2020 ◽  
Vol 14 (2) ◽  
pp. 249-259
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Equations ◽  
Second Order ◽  
General Linear ◽  
Point Transformations

scholarly journals Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Winter Sinkala
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Second Order ◽  
Point Transformation ◽  
Group Method ◽  
Lie Group Method ◽  
Generic Solution ◽  
Point Transformations

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.

scholarly journals Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Sakka Sookmee ◽  
Sergey V. Meleshko
Keyword(s):  
Linear System ◽  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Point Transformations ◽  
Constant Coefficients ◽  
Linearizable System

The necessary form of a linearizable system of two second-order ordinary differential equations y1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

scholarly journals Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
M. Safdar ◽  
Asghar Qadir ◽  
S. Ali
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Canonical Form ◽  
Linear Equations ◽  
Two Dimensions ◽  
Second Order ◽  
Equivalence Classes ◽  
Order Ordinary Differential Equation ◽  
Change Of Variables ◽  
Symmetry Structure

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An “optimal (or simplest) canonical form” of linear systems had been established to obtain the symmetry structure, namely, with 5-, 6-, 7-, 8-, and 15-dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, we provide a “reduced optimal canonical form.” This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6-, 7-, and 15-dimensional algebras for these systems and illustrate our results with examples.

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.

scholarly journals Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complete Classification ◽  
First Integrals ◽  
Second Order ◽  
Point Transformations ◽  
Maximal Algebra ◽  
The Relationship ◽  
Linear Odes

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

4.—Comparison Theorems for a Class of Second-order Non-linear Ordinary Differential Equations

1975 ◽  
Vol 73 ◽  
pp. 77-84
Author(s):  
Donald C. Benson
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Comparison Theorem ◽  
Linear Equations ◽  
Second Order ◽  
Comparison Theorems ◽  
Linear Ordinary Differential Equations ◽  
Second Order Differential Equations ◽  
Liénard Equations ◽  
Fowler Equation

SynopsisIntegral inequalities are used to obtain comparison theorems for a class of second-order differential equations which includes the Emden-Fowler equation, certain Liénard equations, and linear equations of the form d2y/dx2+f(x)y = 0. For these linear equations the results below imply Sturm's classical comparison theorem.

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