| Introduction | |
| Natural Decay and Natural Growth | |
| Differential Equations and Solutions | |
| Mathematical Models and Mathematical Modeling | |
| Scientific Detection of Art Forgery | |
| Basic Concepts and Techniques | |
| A Collection of Mathematical Models | |
| Separable First-Order Equations | |
| Slope Fields | |
| Existence of Unique Solutions | |
| Euler's Method | |
| Runge-Kutta Methods | |
| A Successful Volleyball Serve | |
| Homogeneous Linear Equations | |
| Linear Oscillators | |
| Systems of Linear Algebraic Equations | |
| Theory of Homogeneous Linear Equations | |
| Homogeneous Equations with Constant Coefficients | |
| Real Solutions from Complex Characteristic Values | |
| Multiple Solutions for Repeated Characteristic Values | |
| Some Other Homogeneous Linear Equations | |
| How Long Should Jellyfish Hold their Food? | |
| Nonhomogeneous Linear Equations | |
| More on Linear Oscillator Models | |
| General Solutions for Nonhomogeneous Equations | |
| The Method of Undetermined Coefficients | |
| Forced Linear Oscillators | |
| Solving First-Order Linear Equations | |
| Particular Solutions for Second-Order Equations by Variation of Parameters | |
| A Tuning Circuit for a Radio | |
| Autonomous Equations and Systems | |
| Population Models | |
| The Phase Line | |
| The Phase Plane | |
| The Direction Field and Critical Points | |
| Qualitative Analysis | |
| A Self-Limiting Population | |
| Analytical Methods for Systems | |
| Compartment Models | |
| Eigenvalues and Eigenspaces | |
| Linear Trajectories | |
| Homogeneous Systems with Real Eigenvalues | |
| Homogeneous Systems with Complex Eigenvalues | |
| Additional Solutions for Deficient Matrices | |
| Qualitative Behavior of Nonlinear Systems | |
| Invasion by Disease | |
| The Laplace Transform | |
| Piecewise-Continuous Functions | |
| Definition and Properties of the Laplace Transform | |
| Solution of Initial-Value Problems with the Laplace Transform | |
| Piecewise-Continuous and Impulsive Forcing | |
| Convolution and the Impulse Response Function | |
| Growth of a Structured Population | |
| Vibrating Strings: A Focused Introduction to Partial Differential Equations | |
| Transverse Vibration of a String | |
| The General Solution of the Wave Equation | |
| Vibration Modes of a Finite String | |
| Motion of a Plucked String | |
| Fourier Series | |
| Stringed Instruments and Percussion A Some Additional Topics | |
| Using Integrating Factors to Solve First-Order Linear Equations (Chapter 2) | |
| Proof of the Existence and Uniqueness Theorem for First-Order Equations (Chapter 2) | |
| Error in Numerical Methods (Chapter 2) | |
| Power Series Solutions (Chapter 3) | |
| Matrix Functions (Chapter 6) | |
| Nonhomogeneous Linear Systems (Chapter 6) | |
| The One-Dimensional Heat Equation (Chapter 8) | |
| Laplace's Equation (Chapter 8) | |
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