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Differential Equations: A Modeling Approach

Glenn Ledder
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
, on

As an undergraduate text, this textbook offers a very accessible introduction to ODEs through modeling, using a real-world applied motivation for introducing each major topic in this subject. This becomes a natural and easy way to understand the basics of differential equations through growth and decay scenarios, easily comprehensible oscillations, and more.

In no way does this approach cause the author to cut corners. Instead, greater pains are taken to make sure key ideas can be understood from a direct modeling approach. In between these episodes of modeling explanation, required techniques and theory are covered.

The book extends the expected overview, examples and exercises approach with section-embedded “instant exercises”. These are answered exercises with the answers later in the section instead of the back of the book. Occasional but detailed real world case studies, including an interesting one tracking an art forgery, further help to illuminate concepts. Starting from an introduction to ODEs, the text continues on to include chapters on nonhomogenous linear equations, autonomous equations, systems of differential equations, Laplace transforms, and PDEs.

Tom Schulte (, a graduate student at Oakland University (, has been known to frequent finer sushi establishments.


1 Introduction

1.1 Natural Decay and Natural Growth

1.2 Differential Equations and Solutions

1.3 Mathematical Models and Mathematical Modeling

Case Study 1 Scientific Detection of Art Forgery

2 Basic Concepts and Techniques

2.1 A Collection of Mathematical Models

2.2 Separable First-Order Equations

2.3 Slope Fields

2.4 Existence of Unique Solutions

2.5 Euler's Method

2.6 Runge-Kutta Methods

Case Study 2 A Successful Volleyball Serve

3 Homogeneous Linear Equations

3.1 Linear Oscillators

3.2 Systems of Linear Algebraic Equations

3.3 Theory of Homogeneous Linear Equations

3.4 Homogeneous Equations with Constant Coefficients

3.5 Real Solutions from Complex Characteristic Values

3.6 Multiple Solutions for Repeated Characteristic Values

3.7 Some Other Homogeneous Linear Equations

Case Study 3 How Long Should Jellyfish Hold their Food?

4 Nonhomogeneous Linear Equations

4.1 More on Linear Oscillator Models

4.2 General Solutions for Nonhomogeneous Equations

4.3 The Method of Undetermined Coefficients

4.4 Forced Linear Oscillators

4.5 Solving First-Order Linear Equations

4.6 Particular Solutions for Second-Order Equations by Variation of Parameters

Case Study 4 A Tuning Circuit for a Radio

5 Autonomous Equations and Systems

5.1 Population Models

5.2 The Phase Line

5.3 The Phase Plane

5.4 The Direction Field and Critical Points

5.5 Qualitative Analysis

Case Study 5 A Self-Limiting Population

6 Analytical Methods for Systems

6.1 Compartment Models

6.2 Eigenvalues and Eigenspaces

6.3 Linear Trajectories

6.4 Homogeneous Systems with Real Eigenvalues

6.5 Homogeneous Systems with Complex Eigenvalues

6.6 Additional Solutions for Deficient Matrices

6.7 Qualitative Behavior of Nonlinear Systems

Case Study 6 Invasion by Disease

7 The Laplace Transform

7.1 Piecewise-Continuous Functions

7.2 Definition and Properties of the Laplace Transform

7.3 Solution of Initial-Value Problems with the Laplace Transform

7.4 Piecewise-Continuous and Impulsive Forcing

7.5 Convolution and the Impulse Response Function

Case Study 7 Growth of a Structured Population

8 Vibrating Strings: A Focused Introduction to Partial Differential Equations

8.1 Transverse Vibration of a String

8.2 The General Solution of the Wave Equation

8.3 Vibration Modes of a Finite String

8.4 Motion of a Plucked String

8.5 Fourier Series

Case Study 8 Stringed Instruments and Percussion

A Some Additional Topics

A.1 Using Integrating Factors to Solve First-Order Linear Equations (Chapter 2)

A.2 Proof of the Existence and Uniqueness Theorem for First-Order Equations (Chapter 2)

A.3 Error in Numerical Methods (Chapter 2)

A.4 Power Series Solutions (Chapter 3)

A.5 Matrix Functions (Chapter 6)

A.6 Nonhomogeneous Linear Systems (Chapter 6)

A.7 The One-Dimensional Heat Equation (Chapter 8)

A.8 Laplace's Equation (Chapter 8)