Ledder's innovative, student-centered approach reflects recent research on successful learning by emphasizing connections between new and familiar concepts and by engaging students in a dialogue with the material. Though streamlined, the text is also flexible enough to support a variety of teaching goals, in part through optional topics that give instructors considerable freedom in customizing their courses. Linear algebra is presented in self-contained sections to accommodate both courses that have a linear algebra prerequisite and those that do not. Throughout the text, a wide variety of examples from the physical, life and social sciences, among other areas, are employed to enhance student learning. In-depth Model Problems drawn from everyday experience highlight the key concepts or methods in each section. Other innovative features of the text include Instant Exercises that allow students to quickly test new skills and Case Studies that further explore the powerful problem-solving capability of differential equations. Readers will learn not only how to solve differential equations, but also how to apply their knowledge to areas in mathematics and beyond.
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)
