 
Differential Equations: An Introduction to Modern Methods and Applications, 1st Edition
December 2006, ©2007
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1.1 Some Basic Mathematical Models; Direction Fields.
1.2 Solutions of Some Differential Equations.
1.3 Numerical Approximations: Euler's Method.
1.4 Classification of Differential Equations.
2. First Order Differential Equations.
2.1 Linear Equations; Method of Integrating Factors.
2.2 Separable Equations.
2.3 Modeling with First Order Equations.
2.4 Differences Between Linear and Nonlinear Equations.
2.5 Autonomous Equations and Population Dynamics.
2.6 Exact Equations and Integrating Factors.
2.7 Accuracy of Numerical Methods.
2.8 Improved Euler and RungeKutta Methods.
Summary.
Projects.
2.P.1 Harvesting a Renewable Resource.
2.P.2 Designing a Drip Dispenser for a Hydrology Experiment.
2.P.3 A Mathematical Model of a Groundwater Contaminant Source.
2.P.4 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin.
3. Systems of Two First Order Equations.
3.1 Systems of Two Linear Algebraic Equations.
3.2 Systems of Two First Order Linear Differential Equations.
3.3 Homogeneous Linear Systems with Constant Coefficients.
3.4 Complex Eigenvalues.
3.5 Repeated Eigenvalues.
3.6 A Brief Introduction to Nonlinear Systems.
3.7 Numerical Methods for Systems of First Order Equations.
Summary.
Projects.
3.P.1 EigenvaluePlacement Design of a Satellite Attitude Control System.
3.P.2 Estimating Rate Constants for an Open TwoCompartment Model.
3.P.3 The Ray Theory of Wave Propagation.
3.P.4 A BloodBrain Pharmacokinetic Model.
4. Second Order Linear Equations.
4.1 Definitions and Examples.
4.2 Theory of Second Order Linear Homogeneous Equations.
4.3 Linear Homogeneous Equations with Constant Coefficients.
4.4 Characteristic Equations with Complex Roots.
4.5 Mechanical and Electrical Vibrations.
4.6 Nonhomogeneous Equations; Method of Undetermined Coefficients.
4.7 Forced Vibrations, Frequency Response, and Resonance.
4.8 Variation of Parameters.
Summary.
Projects.
4.P.1 A Vibration Insulation Problem.
4.P.2 Linearization of a Nonlinear Mechanical System.
4.P.3 A SpringMass Event Problem.
4.P.4 Uniformly Distributing Points on a Sphere.
4.P.5 EulerLagrange Equations.
5. The Laplace Transform.
5.1 Definition of the Laplace Transform.
5.2 Properties of the Laplace Transform.
5.3 The Inverse Laplace Transform.
5.4 Solving Differential Equations with Laplace Transforms.
5.5 Discontinuous Functions and Periodic Functions.
5.6 Differential Equations with Discontinuous Forcing Functions.
5.7 Impulse Functions.
5.8 Convolution Integrals and Their Applications.
5.9 Linear Systems and Feedback Control.
Summary.
Projects.
5.P.1 An Electric Circuit Problem.
5.P.2 Effects of Pole Locations on Step Responses of Second Order Systems.
5.P.3 The Watt Governor, Feedback Control, and Stability.
6. Systems of First Order Linear Equations.
6.1 Definitions and Examples.
6.2 Basic Theory of First Order Linear Systems.
6.3 Homogeneous Linear Systems with Constant Coefficients.
6.4 Complex Eigenvalues.
6.5 Fundamental Matrices and the Exponential of a Matrix.
6.6 Nonhomogeneous Linear Systems.
6.7 Defective Matrices.
Summary.
Projects.
6.P.1 A Compartment Model of Heat Flow in a Rod.
6.P.2 Earthquakes and Tall Buildings.
6.P.3 Controlling a SpringMass System to Equilibrium.
7. Nonlinear Differential Equations and Stability.
7.1 Autonomous Systems and Stability.
7.2 Almost Linear Systems.
7.3 Competing Species.
7.4 PredatorPrey Equations.
7.5 Periodic Solutions and Limit Cycles.
7.6 Chaos and Strange Attractors: The Lorenz Equations.
Summary.
Projects.
7.P.1 Modeling of Epidemics.
7.P.2 Harvesting in a Competitive Environment.
7.P.3 The Rossler System.
Appendix A: Matrix Algebra.
A.1 Matrices.
A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank.
A.3 Determinants and Inverses.
A.4 The Eigenvalue Problem.
Appendix B: Complex Variables.
Answers to Selected Problems.
Index.
- Flexible Organization- Organization of chapters, sections, and projects allows for a variety of course configurations depending on desired course goals, topics, and depth of coverage.
- Numerous and Varied Problems- Throughout the text, section exercises of varying levels of difficulty give students hands-on experience in modeling, analysis, and computer experimentation.
- Emphasis on Systems- Systems of first order equations, a central and unifying theme of the text, are introduced early, in Chapter 3, and are used frequently thereafter.
- Linear Algebra and Matrix Methods- Two-dimensional linear algebra sufficient for the study of two first order equations, taken up in Chapter 3, is presented in Section 3.1. Linear algebra and matrix methods required for the study of linear systems of dimension n (Chapter 6) are treated in Appendix A.
- ODE Architect: The companion ODE Architect: provides students with a user- friendly software tool for computing numerical approximations to solutions of systems of differential equations, and for constructing component plots, direction fields, and phase portraits.
- Optional Computing Exercises- In most cases, problems requesting computer generated solutions and graphics are optional.
- Visual Elements- In addition to a large number of illustrations and graphs within the text, physical representations of dynamical systems and animations available in ODE Architect provide students with a strong visual component to the subject.
- Contemporary Project Applications- Optional projects at the end of Chapters 2 through 7 integrate subject matter in the context of exciting, contemporary applications in science and engineering, such as controlling the attitude of a satellite, ray theory of wave propagation, uniformly distributing points on a sphere, and vibration analysis of tall buildings.
- Laplace Transforms- A detailed chapter on
- Control Theory- Ideas and methods from the important application area of control theory are introduced in some examples and projects, and in the last section on Laplace Transforms, all of which are optional.
- Recurring Themes and Applications- Important themes and applications, such as dynamical system formulation, phase portraits, linearization, stability of equilibrium solutions, vibrating systems, and frequency response are revisited and reexamined in different applications and mathematical settings.
- Chapter Summaries- A summary at the end of each chapter provides students and instructors with a birds-eye view of the most important ideas in the chapter.
- Answer to Problems- Answers to all of the problems, many of which are accompanied by a figure, are provided at the end of the book.
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