You are here

Introduction to Difference Equations

Samuel Goldberg
Dover Publications
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a well-written and extremely leisurely introduction to difference equations that includes a wealth of simple applications in the social sciences. The book assumes only high-school algebra and trigonometry as prerequisites. It includes numerous exercises; most of these are drill and are answered by a number or formula, but a few problems in each section explore more advanced concepts. Answers to all problems are in the back of the book. The present publication is a corrected 1986 reprint of a 1958 work.

The book carefully develops all the mathematical background it needs, starting by introducing the concepts of functions and sequences, and proving all the theorems on sequences that would be covered in a careful calculus course. The first major discussion covers the concepts and techniques of finite differences, leading to the observation that many common sequences satisfy simple difference equations. The book then reverses its viewpoint to consider difference equations as the starting point, and proves the existence of sequences satisfying them. The existence theorems are handled in great generality, but after this point the book deals mostly with linear difference equations with constant coefficients, and solves these by using powers of the roots of the auxiliary equation. There is a very thorough treatment of the second-order cases, including the handling of repeated roots and the limiting behavior of solutions to the homogeneous equations (i.e., whether the solution goes to infinity or a finite value, or is oscillatory). The last chapter covers some mathematically more advanced techniques such as stability, generating functions, and Markov processes, but does not go very deeply into these.

The book’s big weakness is that the set of techniques it develops so carefully are not the ones that people use in everyday practice. Working mathematicians and engineers would nearly always use generating functions to solve the types of problems handled here. Generating functions give a uniform treatment of these problems without dividing them into cases based on the roots of the auxiliary equation and the properties of the non-homogeneous part of the equation. Generating functions can even be used in a cookbook fashion analogous to using Laplace transforms to solve differential equations (because of this analogy, engineers use the term Z-transform for the process of forming the generating function). The present book covers these techniques on pp. 189–207, but because it requires some knowledge of calculus it is an optional “starred” section and is not in the mainstream of the development.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  1. Introduction
  2. The Calculus of Finite Differences
    • 1.1 The First Difference Function
    • 1.2 Second and Higher Differences
    • 1.3 The Operator E
    • 1.4 Some Properties of Δ and E
    • 1.5 Equivalence of Operators
    • 1.6 Indefinite Summation : The Operator Δ-1
    • 1.7 Analogies between the Difference and Differential Calculus
  3. Difference Equations
    • 2.1 Basic Definitions
    • 2.2 Solutions of a Difference Equation
    • 2.3 An Existence and Uniqueness Theorem
    • 2.4 The Equation yk+1 = A yk + B
    • 2.5 Sequences
    • 2.6 Solutions as Sequences
    • 2.7 Simple and Compound Interest
    • 2.8 Economic Dynamics
    • 2.9 Inventory Analysis
    • 2.10 A Probability Model for Learning
    • 2.11 Geometric Growth
    • 2.12 Approximating a Differential Equation by a Difference Equation
  4. Linear Difference Equations with Constant Coefficients
    • 3.1 Some Basic Theorems
    • 3.2 Fundamental Sets of Solutions
    • 3.3 General Solution of the Homogeneous Equation
    • 3.4 Particular Solutions of the Complete Equation
    • 3.5 Limiting Behavior of Solutions
    • 3.6 Illustrative Examples from the Social Sciences
    • 3.7 The General Case of Order n
    • 3.8 Linear Differential Equations with Constant Coefficients
  5. Selected Topics
    • 4.1 Equilibrium and Stability
    • 4.2 First-Order Equations and Cobweb Cycles
    • 4.3 A Characteristic-Value Problem
    • 4.4 Generating Functions
    • 4.5 Matrix Methods
  6. Selected References
  7. Answers to Problems
  8. Index