Transforming ordinary linear differential equations to constant coefficient differential equations

Abstract

The main purpose of this thesis is to give a characterization of nth order linear homogeneous differential equations that can be transformed into constant coefficient differential equations. The characterization given makes use of invariance theory originated by G. H. Halphen around 1880. Chapter 1 gives an introduction to the problem of transforming ordinary linear homogeneous differential equations into constant coefficient differential equations. Chapter 1 includes an example as well as enough of the theory of invariants to give a proof of the main theorem of this thesis. Chapter 3 is a development of the transform equations that we make use of throughout this thesis. That is, we make changes of the dependent variable and/or independent variable of a given differential equation. The transformed equation is expressed in terms of the coefficients of the original differential equation and the functions used to define the changes of variables that we have made. In the last section of Chapter 4, we prove an important invariance relation that is used in Chapter 5. The rest of Chapter 4 contains invariance results that are of historical interest as well as being a prelude to the results of Chapter 5. Chapter 5, the most important chapter of this thesis, concerns the invariance theory of Halphen. This invariance theory is needed to give the characterization of nth order differential equations that can be transformed into constant coefficient differential equations. Chapter 6 is devoted to applying the preceding theory to the solution of differential equations that can be transformed into constant coefficient differential equations. The appendix contains some interchange of summation formulas that we use throughout the thesis.