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For the past half century many introductory differential equations courses for science and engineering students have emphasized the formal solution of standard types of differential equations using a (seeming) grab-bag of mechanical solution techniques. The evolution of the present text is based on experience teaching a new course with a greater emphasis on conceptual ideas and the use of computer lab projects to involve students in more intense and sustained problem-solving experiences. Both the conceptual and the computational aspects of such a course depend heavily on the perspective and techniques of linear algebra. Consequently, the study of differential equations and linear algebra in tandem reinforces the learning of both subjects. In this book we have therefore combined core topics in elementary differential equations with those concepts and methods of elementary linear algebra that are needed for a contemporary introduction to differential equations.
The availability of technical computing environments like Maple, Mathematica, and MATLAB is reshaping the current role and applications of differential equations in science and engineering, and has shaped our approach in this text. New technology motivates a shift in emphasis from traditional manual methods to both qualitative and computer-based methods that
The following features of this text are intended to support a contemporary differential equations course with linear algebra that augments traditional core skills with conceptual perspectives:
The following features highlight the computational flavor that distinguishes much of our exposition.
To sample the range of applications in this text, take a look at the following questions:
Organization and Content
The organization and content of the book may be outlined as follows:
Problems, Projects, and the Web Site
About 300 of the text's over 2000 problems are new for this book; the older problems are retained either from Edwards & Penney, Differential Equations: Computing and Modeling (Prentice Hall, 2000) or from our Elementary Linear Algebra (Prentice Hall, 1988). Each section contains a wide variety of computational exercises plus an ample number of applied or conceptual problems.
The answer section includes the answers to most odd-numbered problems and to some of the even-numbered ones. The Student Solutions Manual accompanying this book provides worked-out solutions for most of the odd-numbered problems in the book, while the Instructor's Solutions Manual provides solutions for most of the even-numbered problems as well.
The approximately 30 project sections in the text contain much additional and extended problem material designed to engage students in the exploration and application of computational technology. Most of these projects are expanded considerably in the Computing Projects Manual that accompanies the text and supplements it with additional and sometimes more challenging investigations. Each project section in this manual has parallel Using Maple, Using Mathematics, and Using MATLAB subsections that detail the applicable methods and techniques of each system, and will afford student users an opportunity to compare the merits and styles of different computational systems.
1. First-Order Differential Equations.
Differential Equations and Mathematical Model. Integrals as General and Particular Solutions. Direction Fields and Solution Curves. Separable Equations and Applications. Linear First-Order Equations. Substitution Methods and Exact Equations.
2. Mathematical Models and Numerical Methods.
Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method. The Runge-Kutta Method.
3. Linear Systems and Matrices.
Introduction to Linear Systems. Matrices and Gaussian Elimination. Reduced Row-Echelon Matrices. Matrix Operations. Inverses of Matrices. Determinants. Linear Equations and Curve Fitting.
4. Vector Spaces.
The Vector Space R^3. The Vector Space R
5. Linear Equations of Higher Order.
Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Undetermined Coefficients and Variation of Parameters. Forced Oscillations and Resonance.
6. Eigenvalues and Eigenvectors.
Introduction to Eigenvalues. Diagonalization of Matrices. Applications Involving Powers of Matrices.
7. Linear Systems of Differential Equations.
First-Order Systems and Applications. Matrices and Linear Systems. The Eigenvalue Method for Linear Systems. Second-Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Numerical Methods for Systems.
8. Matrix Exponential Methods.
Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems. Spectral Decomposition Methods.
9. Nonlinear Systems and Phenomena.
Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems.
10. Laplace Transform Methods.
Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions.
11. Power Series Methods.
Introduction and Review of Power Series. Power Series Solutions. Frobenius Series Solutions. Bessel's Equation.
References for Further Study.
Appendix A: The Existence and Uniqueness of Solutions.
Appendix B: Theory of Determinants.
Answers to Selected Problems.
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