Introductory Differential Equations

Introductory Differential Equations, Fourth Edition, offers both narrative explanations and robust sample problems for a first semester course in introductory ordinary differential equations (including Laplace transforms) and a second course in Fourier series and boundary value problems. The book provides the foundations to assist students in learning not only how to read and understand differential equations, but also how to read technical material in more advanced texts as they progress through their studies.

This text is for courses that are typically called (Introductory) Differential Equations, (Introductory) Partial Differential Equations, Applied Mathematics, and Fourier Series. It follows a traditional approach and includes ancillaries like Differential Equations with Mathematica and/or Differential Equations with Maple. Because many students need a lot of pencil-and-paper practice to master the essential concepts, the exercise sets are particularly comprehensive with a wide array of exercises ranging from straightforward to challenging. There are also new applications and extended projects made relevant to everyday life through the use of examples in a broad range of contexts.

This book will be of interest to undergraduates in math, biology, chemistry, economics, environmental sciences, physics, computer science and engineering.

• Provides the foundations to assist students in learning how to read and understand the subject, but also helps students in learning how to read technical material in more advanced texts as they progress through their studies
• Exercise sets are particularly comprehensive with a wide range of exercises ranging from straightforward to challenging
• Includes new applications and extended projects made relevant to "everyday life" through the use of examples in a broad range of contexts
• Accessible approach with applied examples and will be good for non-math students, as well as for undergrad classes

Undergraduates in math, biology, chemistry, economics, environmental sciences, physics, computer science and engineering

Preface

• Technology
• Applications
• Style
• Features
• Pedagogical Features
• Content

Chapter 1: Introduction to Differential Equations

• Abstract
• 1.1 Introduction to Differential Equations: Vocabulary
• Exercises 1.1
• 1.2 A Graphical Approach to Solutions: Slope Fields and Direction Fields
• Exercises 1.2
• Chapter 1 Summary: Essential Concepts and Formulas
• Chapter 1 Review Exercises

Chapter 2: First-Order Equations

• Abstract
• 2.1 Introduction to First-Order Equations
• Exercises 2.1
• 2.2 Separable Equations
• Exercises 2.2
• 2.3 First-Order Linear Equations
• Exercises 2.3
• 2.4 Exact Differential Equations
• Exercises 2.4
• 2.5 Substitution Methods and Special Equations
• Exercises 2.5
• 2.6 Numerical Methods for First-Order Equations
• Exercises 2.6
• Chapter 2 Summary: Essential Concepts and Formulas
• Chapter 2 Review Exercises
• Differential Equations at Work

Chapter 3: Applications of First-Order Differential Equations

• Abstract
• 3.1 Population Growth and Decay
• Exercises 3.1
• 3.2 Newton’s Law of Cooling and Related Problems
• Exercises 3.2
• 3.3 Free-Falling Bodies
• Exercises 3.3
• Chapter 3 Summary: Essential Concepts and Formulas
• Chapter 3 Review Exercises
• Differential Equations at Work

Chapter 4: Higher Order Equations

• Abstract
• 4.1 Second-Order Equations: An Introduction
• Exercises 4.1
• 4.2 Solutions of Second-Order Linear Homogeneous Equations with Constant Coefficients
• Exercises 4.2
• 4.3 Solving Second-Order Linear Equations: Undetermined Coefficients
• Exercises 4.3
• 4.4 Solving Second-Order Linear Equations: Variation of Parameters
• Exercises 4.4
• 4.5 Solving Higher Order Linear Homogeneous Equations
• Exercises 4.5
• 4.6 Solving Higher Order Linear Equations: Undetermined Coefficients and Variation of Parameters
• Exercises 4.6
• 4.7 Cauchy-Euler Equations
• Exercises 4.7
• 4.8 Power Series Solutions of Ordinary Differential Equations
• Exercises 4.8
• 4.9 Series Solutions of Ordinary Differential Equations
• Exercises 4.9
• Chapter 4 Summary: Essential Concepts and Formulas
• Chapter 4 Review Exercises
• Differential Equations at Work

Chapter 5: Applications of Higher Order Differential Equations

• Abstract
• 5.1 Simple Harmonic Motion
• 5.2 Damped Motion
• 5.3 Forced Motion
• 5.4 Other Applications
• 5.5 The Pendulum Problem
• Chapter 5 Summary: Essential Concepts and Formulas
• Chapter 5 Review Exercises
• Differential Equations at Work

Chapter 6: Systems of Differential Equations

• Abstract
• 6.1 Introduction
• Exercises 6.1
• 6.2 Review of Matrix Algebra and Calculus
• Exercises 6.2
• 6.3 An Introduction to Linear Systems
• Exercises 6.3
• 6.4 First-Order Linear Homogeneous Systems With Constant Coefficients
• Exercises 6.4
• 6.5 First-Order Linear Nonhomogeneous Systems: Undetermined Coefficients and Variation of Parameters
• Exercises 6.5
• 6.6 Phase Portraits
• Exercises 6.6
• 6.7 Nonlinear Systems
• Exercises 6.7
• 6.8 Numerical Methods
• Exercises 6.8
• Chapter 6 Summary: Essential Concepts and Formulas
• Chapter 6 Review Exercises
• Differential Equations at Work

Chapter 7: Applications of Systems of Ordinary Differential Equations

• Abstract
• 7.1 Mechanical and Electrical Problems With First-Order Linear Systems
• Exercises 7.1
• 7.2 Diffusion and Population Problems With First-Order Linear Systems
• Exercises 7.2
• 7.3 Nonlinear Systems of Equations
• Exercises 7.3
• Chapter 7 Summary: Essential Concepts and Formulas
• Chapter 7 Review Exercises
• Differential Equations at Work

Chapter 8: Introduction to the Laplace Transform

• Abstract
• 8.1 The Laplace Transform: Preliminary Definitions and Notation
• Exercises 8.1
• 8.2 The Inverse Laplace Transform
• Exercises 8.2
• 8.3 Solving Initial-Value Problems with the Laplace Transform
• Exercises 8.3
• 8.4 Laplace Transforms Of Several Important Functions
• Exercises 8.4
• 8.5 The Convolution Theorem
• Exercises 8.5
• 8.6 Laplace Transform Methods for Solving Systems
• Exercises 8.6
• 8.7 Some Applications Using Laplace Transforms
• Chapter 8 Review Exercises
• Differential Equations at Work

Answers to Selected Exercises

• Exercises 1.1
• Exercises 1.2
• Chapter 1 Review Exercises
• Exercises 2.1
• Exercises 2.2
• Exercises 2.3
• Exercises 2.4
• Exercises 2.5
• Exercises 2.6
• Chapter 2 Review Exercises
• Exercises 3.1
• Exercises 3.2
• Exercises 3.3
• Chapter 3 Review Exercises
• Exercises 4.1
• Exercises 4.2
• Exercises 4.3
• Exercises 4.4
• Exercises 4.5
• Exercises 4.6
• Exercises 4.7
• Exercises 4.8
• Exercises 4.9
• Chapter 4 Review Exercises
• Exercises 5.1
• Exercises 5.2
• Exercises 5.3
• Exercises 5.4
• Exercises 5.5
• Chapter 5 Review Exercises
• Exercises 6.1
• Exercises 6.2
• Exercises 6.3
• Exercises 6.4
• Exercises 6.5
• Exercises 6.6
• Exercises 6.7
• Exercises 6.8
• Chapter 6 Review Exercises
• Exercises 7.1
• Exercises 7.2
• Exercises 7.3
• Chapter 7 Review Exercises
• Exercises 8.1
• Exercises 8.2
• Exercises 8.3
• Exercises 8.4
• Exercises 8.5
• Exercises 8.6
• Exercises 8.7
• Chapter 8 Review ExercisesIndex