Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra

J. C. Ndogmo

doi:10.1155/2016/1796316

Abstract

An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution.

Highlights

Linear ordinary differential equations are quite probably the most common type of differential equations that occur in physics and in many other mathematically based fields of science
The results obtained far in this paper will be used to show amongst others that lodes of maximal symmetry are highly solvable and that it is the case for the second-order source equation (18) whose solutions completely determine those of the corresponding lodes of maximal symmetry
On one hand and on the basis of the results obtained in the previous section, it follows from Theorem 3 that can we generate a lode of maximal symmetry of any order and for any nonzero function r, and the general solution of any such equation is given in closed form in terms of the source parameter r by (33)

Summary

Introduction

Linear ordinary differential equations (lodes) are quite probably the most common type of differential equations that occur in physics and in many other mathematically based fields of science. That short paper left a number of important questions unanswered It does not provide, for instance, any expression for the point transformation mapping a given equation of maximal symmetry to the canonical form. We provide a much simpler differential operator than that found in [3] for generating linear iterative equations of a general order This gives rise to a simple algorithm for testing lodes for maximal symmetry based solely on their coefficients. In contrast to the very well-known paper by Ermakov [11] who managed to find only some very specific cases from a restricted class for which the secondorder source equation is solvable, we provide large families of second-order equations for which the general solution is given by simple algebraic formulas. All such families are parameterized by an entirely arbitrary nonzero function and the general solutions found for the second-order source equation yield through a very simple quadratic formula that for the whole corresponding class of equations of maximal symmetry of a general order

Iterations of Linear Equations

Reduction to Canonical Form and General Solution

Solvability of Equations of Maximal Symmetry

Concluding Remarks