Ordinary Differential Equations

William Cox

• Publisher: Butterworth-Heinemann
• Year: 1995
• ISBN:
  9780080928678
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.
The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further study of partial differential equations.

1

Introduction and a Look Ahead

1.1 Getting an overview

Coming to a new mathematics topic such as differential equations is like entering a new land covered with forests of details, conceptual peaks, rivers and highways of understanding, exquisite hidden valleys of interest, and broad featureless plains of monotony. There are various ways you can explore a new land: a guided tour with an expert; a ‘follow the colour coded signposts’ forest walk; a leisurely cruise along the main highways, taking in the sights; live there for a while, wandering through the highways and byways, climbing the peaks by different routes. But however you do it, before you delve into the guide book, you will greatly benefit from a preliminary study of the atlas, to get your bearings. The problem with this geographical analogy is that whilst we are already familiar with the geographical and topographical terminology needed to describe the features of a new land, this is rarely the case in a new mathematics topic – it is doubly difficult to distinguish the wood from the trees if you don’t know what either looks like! However, with some imagination we can lay out the broad features of a new topic using already familiar ideas – and that’s what we will do in this chapter, relying only on very elementary knowledge of calculus and differential equations.

The study of differential equations can be split into a number of broad areas

• Notation and terminology

– What is the order of a differential equation?

– What is a ‘solution’ to a differential equation?

• Analytical aspects

– Existence theorems – under what conditions does a solution exist?

– Uniqueness theorems – when is a solution unique?

• Methods of solution

– Range of techniques and their applicability. Exact and approximate methods.

• Applications

– Constructing and solving models with differential equations.

Depending on your interests, some of these areas may be more relevant than others to you. For example the student of pure mathematics might reasonably be expected to use and prove an existence theorem for a particular class of equations. An engineer on the other hand might be more interested in solution methods (particularly numerical methods).

The remaining sections of this chapter enable you to develop an overview of the subject matter and structure of the rest of the book, whatever your interests. The individual chapters will give the details and provide opportunity to hone the skills which you need.

1.2 Notation, terminology and analytical aspects

Before we can start to discuss differential equations there is a certain amount of notation and terminology to deal with. The order and degree of ordinary differential equations are used to classify equations, usually for the purposes of analysing or solving them, There is a major division between linear and nonlinear equations. With a few simple exceptions...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Dedication
• Copyright
• Series Preface
• Preface
• Acknowledgements
• Chapter 1: Introduction and a Look Ahead
• Chapter 2: First-order Differential Equations – Methods and Models
• Chapter 3: First-order Differential Equations – Analysis and Approximation
• Chapter 4: Second- and Higher-order Homogenous Linear Differential Equations
• Chapter 5: Inhomogeneous Linear Differential Equations
• Chapter 6: Laplace Transform Methods for Solving Initial Value Problems
• Chapter 7: Systems of Linear Differential Equations
• Chapter 8: Series Solution of Linear Differential Equations
• Chapter 9: Special Functions and Orthogonal Expansions
• Chapter 10: An Introduction to Nonlinear Systems
• Appendix – Chapter Summaries
• Answers to Exercises
• Bibliography
• Index