Symmetry and transformation properties of linear iterative ordinary differential equation

Abstract

Solutions of linear iterative equations and expressions for these solutions in terms of the parameters of the source equation are obtained. Based on certain properties of iterative equations, nding the solutions is reduced to nding group-invariant solutions of the second-order source equation. We have therefore found classes of solutions to the source equations. Regarding the expressions of the solutions in terms of the parameters of the source equation, an ansatz is made on the original parameters r and s, by letting them be functions of a speci c type such as monomials, functions of exponential and logarithmic type. We have also obtained an expression for the source parameters of the transformed equation under equivalence transformations and we have looked for the conservation laws of the source equation. We conducted this work with a special emphasis on second-, third- and fourth-order equations, although some of our results are valid for equations of a general order.