Differential Equations: A Modeling Approach

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.

Table of Contents:
1 INTRODUCTION 1. Differential Equations and Solutions 2. Mathematical Models and Mathematical Modeling 3. Case Study---A Successful Volleyball Serve 2 BASIC TECHNIQUES AND CONCEPTS 1. Natural Decay 2. Separable First-Order Equations 3. Slope Fields 4. Euler's Method 5. Error and the Modified Euler Method 6. Existence of Unique Solutions 7. Case Study---Scientific Detection of Art Forgery 3 HOMOGENEOUS LINEAR EQUATIONS 1. Linear Oscillators 2. Linearity 3. Homogeneous Equations with Constant Coefficients 4. Real Solutions from Complex Characteristic Values 5. Multiple Solutions for Multiple Characteristic Values 6. Other Homogeneous Linear Equations 7. Case Study---Chemotherapy for Cancer Treatment 4 NONHOMOGENEOUS LINEAR EQUATIONS 1. General Solutions for Nonhomogeneous Equations 2. The Method of Undetermined Coefficients 3. Forced Linear Oscillators 4. Solution of First-Order Linear Equations 5. Particular Solutions by Variation of Parameters 6. Case Study---A Tuning Circuit for a Radio 5 GRAPHICAL METHODS FOR AUTONOMOUS PROBLEMS 1. Population Models 2. The Phase Line 3. The Phase Plane 4. Equilibrium Points and Stability 5. Qualitative Analysis 6. Case Study---A Self-Limiting Population 6 SYSTEMS OF LINEAR EQUATIONS 1. A Linear Algebra Primer 2. Straight-Line Solutions 3. Systems with Real Eigenvalues 4. Systems with Complex Eigenvalues 5. Additional Solutions for Multiple Eigenvalues 6. Local Analysis of Critical Points for Nonlinear Systems 7. Case Study---Dynamics of Contagious Diseases 7 THE LAPLACE TRANSFORM 1. More on Linear Oscillator Models 2. Definition and Properties of the Laplace Transform 3. Solution of Initial Value Problems with the Laplace Transform 4. Differential Equations with Discontinuous or Impulsive Forcing 5. Case Study---TBA 8 VIBRATING STRINGS AND TRAVELING WAVES 1. Transverse Vibration of a String 2. Vibration Modes of a Finite String 3. Motion of a Plucked String 4. Case Study---TBA SOME ADDITIONAL TOPICS 1. Using Integrating Factors to Solve First-Order Linear Equations 2. Runge-Kutta Methods 3. Proof of the Existence and Uniqueness Theorem for First Order Equations 4. Power Series Solutions 5. Nonautonomous Dynamical Systems 6. Fundamental Matrices 7. Nonhomogeneous Linear Systems 8. Regular Perturbation 9. Projectiles in Sports