Differential equations : a modeling approach / Glenn Ledder.

Includes index.

1 Introduction1.1 Natural Decay and Natural Growth1.2 Differential Equations and Solutions1.3 Mathematical Models and Mathematical ModelingCase Study 1 Scientific Detection of Art Forgery2 Basic Concepts and Techniques2.1 A Collection of Mathematical Models2.2 Separable First-Order Equations2.3 Slope Fields2.4 Existence of Unique Solutions2.5 Euler's Method2.6 Runge-Kutta MethodsCase Study 2 A Successful Volleyball Serve3 Homogeneous Linear Equations3.1 Linear Oscillators3.2 Systems of Linear Algebraic Equations3.3 Theory of Homogeneous Linear Equations3.4 Homogeneous Equations with Constant Coefficients3.5 Real Solutions from Complex Characteristic Values3.6 Multiple Solutions for Repeated Characteristic Values3.7 Some Other Homogeneous Linear EquationsCase Study 3 How Long Should Jellyfish Hold their Food?4 Nonhomogeneous Linear Equations4.1 More on Linear Oscillator Models4.2 General Solutions for Nonhomogeneous Equations4.3 The Method of Undetermined Coefficients4.4 Forced Linear Oscillators4.5 Solving First-Order Linear Equations4.6 Particular Solutions for Second-Order Equations by Variation of ParametersCase Study 4 A Tuning Circuit for a Radio5 Autonomous Equations and Systems5.1 Population Models5.2 The Phase Line5.3 The Phase Plane5.4 The Direction Field and Critical Points5.5 Qualitative AnalysisCase Study 5 A Self-Limiting Population6 Analytical Methods for Systems6.1 Compartment Models6.2 Eigenvalues and Eigenspaces6.3 Linear Trajectories6.4 Homogeneous Systems with Real Eigenvalues6.5 Homogeneous Systems with Complex Eigenvalues6.6 Additional Solutions for Deficient Matrices6.7 Qualitative Behavior of Nonlinear SystemsCase Study 6 Invasion by Disease7 The Laplace Transform7.1 Piecewise-Continuous Functions7.2 Definition and Properties of the Laplace Transform7.3 Solution of Initial-Value Problems with the Laplace Transform7.4 Piecewise-Continuous and Impulsive Forcing7.5 Convolution and the Impulse Response FunctionCase Study 7 Growth of a Structured Population8 Vibrating Strings: A Focused Introduction to Partial Differential Equations8.1 Transverse Vibration of a String8.2 The General Solution of the Wave Equation8.3 Vibration Modes of a Finite String8.4 Motion of a Plucked String8.5 Fourier SeriesCase Study 8 Stringed Instruments and PercussionA Some Additional TopicsA.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)