### Synopses & Reviews

This corrected third printing retains the authors'main emphasis on ordinary differential equations. It is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. The authors have taken the view that a differential equations theory defines functions; the object of the theory is to understand the behaviour of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods, and the companion software, MacMath, is designed to bring these notions to life.

#### Review

.H. Hubbard and B.H. West Differential Equations: A Dynamical Systems Approach "As attention has moved from idealized linear differential equations to the nonlinear equations of the real world, there has been a concomitant change of emphasis, even a paradigm shift, from quantitative methods, analytical and numerical, to qualitative methods. Though qualitative methods originated with Poincaré (Poincare) early in this century, the era of large-scale computation and computer graphics has spurred development through a profound confluence of theory and experiment. Hubbard and West demonstrate convincingly that it is feasible and necessary to incorporate this sophisticated viewpoint into the most elementary stages of differential equations pedagogy, even where the focus is still on linear equations. This is an unusual book in being both rigorous and squarely addressed to those students with the most practical needs. The well known exchange value between pictures and words is thoroughly exploited to ground fundamental concepts."--CHOICE

#### Review

.H. Hubbard and B.H. West

*Differential Equations: A Dynamical Systems Approach*

*"As attention has moved from idealized linear differential equations to the nonlinear equations of the real world, there has been a concomitant change of emphasis, even a paradigm shift, from quantitative methods, analytical and numerical, to qualitative methods. Though qualitative methods originated with Poincaré (Poincare) early in this century, the era of large-scale computation and computer graphics has spurred development through a profound confluence of theory and experiment. Hubbard and West demonstrate convincingly that it is feasible and necessary to incorporate this sophisticated viewpoint into the most elementary stages of differential equations pedagogy, even where the focus is still on linear equations. This is an unusual book in being both rigorous and squarely addressed to those students with the most practical needs. The well known exchange value between pictures and words is thoroughly exploited to ground fundamental concepts."--*CHOICE

#### Synopsis

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM) . The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface Consider a first order differential equation of form x' = f ( t, x). In elemen tary courses one frequently gets the impression that such equations can usually be "solved," i. e., that explicit formulas for the solutions (in terms of powers, exponentials, trigonometric functions, and the like) can usually be found. Nothing could be further from the truth."

#### Synopsis

This corrected third printing retains the authors'main emphasis on ordinary differential equations. It is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. The authors have taken the view that a differential equations theory defines functions; the object of the theory is to understand the behaviour of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods, and the companion software, MacMath, is designed to bring these notions to life.

### Table of Contents

Contents Series Preface for Texts in Applied Mathematics Preface Acknowledgments Ways to Use This Book Introduction Chapter 1 Qualitative Methods 1.1 Fields of Slopes and Sketching of Solutions 1.2 Qualitative Description of Solutions 1.3 Fences 1.4 Funnels and Antifunnels 1.5 The Use of Fences, Funnels, and Antifunnels Exercises Chapter 2 Analystic Methods 2.1 Separation of Variables 2.2 Linear Differential Equations of First Order 2.3 Variation of Parameters 2.4 Bank Accounts and Linear Differential Equations