Elementary Differential Equations and Boundary Value Problems, Student Solutions Manual

This is the Student Solutions Manual to accompany Elementary Differential Equations, 11th Edition. Elementary Differential Equations, 11th Edition is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two] or three] semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.

Preface vii Chapter 1 Introduction 1 1.1 Some Basic Mathematical Models; Direction Fields 1 1.2 Solutions of Some Differential Equations 9 1.3 Classification of Differential Equations 17 1.4 Historical Remarks 23 Chapter 2 First Order Differential Equations 29 2.1 Linear Equations with Variable Coefficients 29 2.2 Separable Equations 40 2.3 Modeling with First Order Equations 47 2.4 Differences Between Linear and Nonlinear Equations 64 2.5 Autonomous Equations and Population Dynamics 74 2.6 Exact Equations and Integrating Factors 89 2.7 Numerical Approximations: Euler's Method 96 2.8 The Existence and Uniqueness Theorem 105 2.9 First Order Difference Equations 115 Chapter 3 Second Order Linear Equations 129 3.1 Homogeneous Equations with Constant Coefficients 129 3.2 Fundamental Solutions of Linear Homogeneous Equations 137 3.3 Linear Independence and the Wronskian 147 3.4 Complex Roots of the Characteristic Equation 153 3.5 Repeated Roots; Reduction of Order 160 3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients 169 3.7 Variation of Parameters 179 3.8 Mechanical and Electrical Vibrations 186 3.9 Forced Vibrations 200 Chapter 4 Higher Order Linear Equations 209 4.1 General Theory of nth Order Linear Equations 209 4.2 Homogeneous Equations with Constant Coeffients 214 4.3 The Method of Undetermined Coefficients 222 4.4 The Method of Variation of Parameters 226 Chapter 5 Series Solutions of Second Order Linear Equations 231 5.1 Review of Power Series 231 5.2 Series Solutions near an Ordinary Point, Part I 238 5.3 Series Solutions near an Ordinary Point, Part II 249 5.4 Regular Singular Points 255 5.5 Euler Equations 260 5.6 Series Solutions near a Regular Singular Point, Part I 267 5.7 Series Solutions near a Regular Singular Point, Part II 272 5.8 Bessel's Equation 280 Chapter 6 The Laplace Transform 293 6.1 Definition of the Laplace Transform 293 6.2 Solution of Initial Value Problems 299 6.3 Step Functions 310 6.4 Differential Equations with Discontinuous Forcing Functions 317 6.5 Impulse Functions 324 6.6 The Convolution Integral 330 Chapter 7 Systems of First Order Linear Equations 339 7.1 Introduction 339 7.2 Review of Matrices 348 7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 357 7.4 Basic Theory of Systems of First Order Linear Equations 368 7.5 Homogeneous Linear Systems with Constant Coefficients 373 7.6 Complex Eigenvalues 384 7.7 Fundamental Matrices 393 7.8 Repeated Eigenvalues 401 7.9 Nonhomogeneous Linear Systems 411 Chapter 8 Numerical Methods 419 8.1 The Euler or Tangent Line Method 419 8.2 Improvements on the Euler Method 430 8.3 The Runge-Kutta Method 435 8.4 Multistep Methods 439 8.5 More on Errors; Stability 445 8.6 Systems of First Order Equations 455 Chapter 9 Nonlinear Differential Equations and Stability 459 9.1 The Phase Plane; Linear Systems 459 9.2 Autonomous Systems and Stability 471 9.3 Almost Linear Systems 479 9.4 Competing Species 491 9.5 Predator-Prey Equations 503 9.6 Liapunov's Second Method 511 9.7 Periodic Solutions and Limit Cycles 521 9.8 Chaos and Strange Attractors; the Lorenz Equations 532 Chapter 10 Partial Differential Equations and Fourier Series 541 10.1 Two-Point Boundary Valve Problems 541 10.2 Fourier Series 547 10.3 The Fourier Convergence Theorem 558 10.4 Even and Odd Functions 564 10.5 Separation of Variables; Heat Conduction in a Rod 573 10.6 Other Heat Conduction Problems 581 10.7 The Wave Equation; Vibrations of an Elastic String 591 10.8 Laplace's Equation 604 Appendix A. Derivation of the Heat Conduction Equation 614 Appendix B. Derivation of the Wave Equation 617 Chapter 11 Boundary Value Problems and Sturm-Liouville Theory 621 11.1 The Occurrence of Two Point Boundary Value Problems 621 11.2 Sturm-Liouville Boundary Value Problems 629 11.3 Nonhomogeneous Boundary Value Problems 641 11.4 Singular Sturm-Liouville Problems 656 11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 663 11.6 Series of Orthogonal Functions: Mean Convergence 669 Answers to Problems 679 Index 737