A Modern Introduction to Differential Equations

A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations. Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering. This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful. - student friendly readability- assessible to the average student - early introduction of qualitative and numerical methods - large number of exercises taken from biology, chemistry, economics, physics and engineering - Exercises are labeled depending on difficulty/sophistication - Full ancillary package including; Instructors guide, student solutions manual and course management system - end of chapter summaries - group projects Front Cover......Page 1 A Modern Introduction to Differential Equations......Page 4 Copyright Page......Page 5 Dedication......Page 6 Table of Contents......Page 8 Preface......Page 12 Acknowledgments......Page 16 Introduction......Page 18 1.1 Basic Terminology......Page 19 1.2 Solutions of Differential Equations......Page 25 1.3 Initial-Value Problems and Boundary-Value Problems......Page 30 Summary......Page 42 Project 1-1......Page 43 Introduction......Page 44 2.1 Separable Equations......Page 45 2.2 Linear Equations......Page 55 2.3 Compartment Problems......Page 65 2.4 Slope Fields......Page 73 2.5 Phase Lines and Phase Portraits......Page 85 2.6 Equilibrium Points: Sinks, Sources, and Nodes......Page 91 *2.7 Bifurcations......Page 98 *2.8 Existence and Uniqueness of Solutions......Page 105 Summary......Page 112 Project 2-1......Page 113 Project 2-2......Page 114 3.1 Euler’S Method......Page 116 3.2 The Improved Euler Method......Page 135 3.3 More Sophisticated Numerical Methods: Runge-Kutta and Others......Page 139 Summary......Page 144 Project 3-1......Page 146 4.1 Homogeneous Second-Order Linear Equations with Constant Coefficients......Page 148 4.2 Nonhomogeneous Second-Order Linear Equations with Constant Coefficients......Page 158 4.3 The Method of Undetermined Coefficients......Page 161 4.4 Variation of Parameters......Page 169 4.5 Higher-Order Linear Equations with Constant Coefficients......Page 176 4.6 Higher-Order Equations and Their Equivalent Systems......Page 181 4.7 The Qualitative Analysis of Autonomous Systems......Page 190 4.8 Spring-Mass Problems......Page 205 *4.9 Existence and Uniqueness......Page 220 4.10 Numerical Solutions......Page 224 Summary......Page 233 Project 4-1......Page 236 5.1 Systems and Matrices......Page 238 5.2 Two-Dimensional Systems of First-Order Linear Equations......Page 244 5.3 The Stability of Homogeneous Linear Systems: Unequal Real Eigenvalues......Page 259 5.4 The Stability of Homogeneous Linear Systems: Equal Real Eigenvalues......Page 271 5.5 The Stability of Homogeneous Linear Systems: Complex Eigenvalues......Page 278 5.6 Nonhomogeneous Systems......Page 287 5.7 Generalizations :The n × n Case (n ≥ 3)......Page 298 Summary......Page 316 Project 5-1......Page 318 Project 5-2......Page 319 Introduction......Page 320 6.1 The Laplace Transform of Some Important Functions......Page 321 6.2 The Inverse Transform and The Convolution......Page 328 6.3 Transforms of Discontinuous Functions......Page 340 6.4 Transforms of Impulse Functions—The Dirac Delta Function......Page 348 6.5 Transforms of Systems of Linear Differential Equations......Page 353 6.6 A Qualitative Analysis Via The Laplace Transform......Page 358 Summary......Page 366 Project 6-1......Page 368 7.1 Equilibria of Nonlinear Systems......Page 370 7.2 Linear Approximation at Equilibrium Points......Page 375 7.3 The Poincaré-Lyapunov Theorem......Page 384 7.4 Two Important Examples of Nonlinear Equations and Systems......Page 392 *7.5 Van Der Pol’S Equation and Limit Cycles......Page 402 Summary......Page 411 Project 7-1......Page 413 A.1 Local Linearity: The Tangent Line Approximation......Page 416 A.3 The Taylor Polynomial/Taylor Series......Page 417 A.4 The Fundamental Theorem Of Calculus (FTC)......Page 420 A.5 Partial Fractions......Page 421 A.6 Improper Integrals......Page 422 A.7 Functions of Several Variables/Partial Derivatives......Page 425 A.8 The Tangent Plane: The Taylor Expansion of F(X,Y)......Page 426 B.1 Vectors and Vector Algebra; Polar Coordinates......Page 428 B.2 Matrices and Basic Matrix Algebra......Page 431 B.3 Linear Transformations and Matrix Multiplication......Page 432 B.4 Eigenvalues and Eigenvectors......Page 438 C.1 Complex Numbers: The Algebraic View......Page 442 C.2 Complex Numbers: The Geometric View......Page 444 C.4 Euler’S Formula......Page 445 D.1 Power Series Solutions of First-Order Equations......Page 446 D.2 Series Solutions of Second-Order Linear Equations: Ordinary Points......Page 448 D.3 Regular Singular Points: The Method of Frobenius......Page 451 D.4 The Point at Infinity......Page 456 D.5 Some Additional Special Differential Equations......Page 457 Answers/Hints to Odd-Numbered Exercises......Page 460 Index......Page 522 "A Modern Introduction to Differential Equations" presents a solid yet highly accessible introduction to differential equations, developing the concepts from a dynamical systems perspective and employing technology to treat topics graphically, numerically and analytically. This text is designed to be appropriate for a wide variety of students and exists as a natural successor to any modern calculus sequence. Ancillary list: * Online ISM- http: //(http://textbooks.elsevier.com/web/product_details.aspx?isbn=9780123747464) textbooks.elsevier.com/web/product_de... * Online SSM- http: //(http://www.elsevierdirect.com/product.jsp?isbn=9780123747464) www.elsevierdirect.com/product.jsp?is... * Algorithmic Testing by Maple- http: //(http://www.elsevierdirect.com/product.jsp?isbn=9780123747464) www.elsevierdirect.com/product.jsp?is... * Sample content, Ebook- http: //(http://www.elsevierdirect.com/product.jsp?isbn=9780123747464) www.elsevierdirect.com/product.jsp?is... * Image collection- http: //(http://www.elsevierdirect.com/companion.jsp?ISBN=9780123747464) www.elsevierdirect.com/companion.jsp?... - student friendly readability- assessible to the average student - early introduction of qualitative and numerical methods - large number of exercises taken from biology, chemistry, economics, physics and engineering - Exercises are labeled depending on difficulty/sophistication - Full ancillary package including; Instructors guide, student solutions manual and course management system - end of chapter summaries - group projects Probability and Statistics are studied by most science students, usually as a second- or third-year course. Many current texts in the area are just cookbooks and, as a result, students do not know why they perform the methods they are taught, or why the methods work. The strength of this book is that it readdresses these shortcomings; by using examples, often from real-life and using real data, the authors can show how the fundamentals of probabilistic and statistical theories arise intuitively. It provides a tried and tested, self-contained course, that can also be used for self-study. A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to the students. In addition the book contains over 350 exercises, half of which have answers, of which half have full solutions. A website at www.springeronline.com/1-85233-896-2 gives access to the data files used in the text, and, for instructors, the remaining solutions. The only pre-requisite for the book is a first course in calculus; the text covers standard statistics and probability material, and develops beyond traditional parametric models to the Poisson process, and on to useful modern methods such as the bootstrap. This will be a key text for undergraduates in Computer Science, Physics, Mathematics, Chemistry, Biology and Business Studies who are studying a mathematical statistics course, and also for more intensive engineering statistics courses for undergraduates in all engineering subjects "A Modern Introduction to Probability and Statistics has numerous quick exercises to give direct feedback to the students. In addition the book contains over 350 exercises, half of which have answers, of which half have full solutions. The only pre-requisite for the book is a first course in calculus; the text covers standard statistics and probability material, and develops beyond traditional parametric models to the Poisson process, and on to useful modern methods such as the bootstrap." "This will be a key text for undergraduates in Computer Science, Physics, Mathematics, Chemistry, Biology and Business Studies who are studying a mathematical statistics course, and also for more intensive engineering statistics courses for undergraduates in all engineering subjects."--Jacket Suitable for self study Use real examples and real data sets that will be familiar to the audience Introduction to the bootstrap is included this is a modern method missing in many other books