SL(3,<i>R</i>) as the group of symmetry transformations for all one-dimensional linear systems

Abstract

The converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order ẍ+f2(t)ẋ+f1(t)x =f0(t). The eight-parameter realizations of the symmetry group are obtained in the form ℱ−1𝒫2 ℱ, where ℱ stands for transformations of (t,x) that depend exclusively on the fundamental solutions of the equation, and where 𝒫2 is an arbitrary projective transformation in the plane. Thus it is shown that the full point symmetry group corresponds to SL(3,R) indeed, without recourse to the Lie algebra. Also, a technique is obtained for calculating the finite point symmetry realization of SL(3,R) for any given one-dimensional linear system. Some miscellaneous examples are given.