Preface   1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models 1.2 Integrals as General and Particular Solutions 1.3 Slope Fields and Solution Curves 1.4 Separable Equations and Applications 1.5 Linear First-Order Equations 1.6 Substitution Methods and Exact Equations 1.7 Population Models 1.8 Acceleration-Velocity Models   2 Linear Equations of Higher Order 2.1 Introduction: Second-Order Linear Equations 2.2 General Solutions of Linear Equations 2.3 Homogeneous Equations with Constant Coefficients 2.4 Mechanical Vibrations 2.5 Nonhomogeneous Equations and Undetermined Coefficients 2.6 Forced Oscillations and Resonance 2.7 Electrical Circuits 2.8 Endpoint Problems and Eigenvalues   3 Power Series Methods 3.1 Introduction and Review of Power Series 3.2 Series Solutions Near Ordinary Points 3.3 Regular Singular Points 3.4 Method of Frobenius: The Exceptional Cases 3.5 Bessel's Equation 3.6 Applications of Bessel Functions   4 LaplaceTransform Methods 4.1 Laplace Transforms and Inverse Transforms 4.2 Transformation of Initial Value Problems 4.3 Translation and Partial Fractions 4.4 Derivatives, Integrals, and Products of Transforms 4.5 Periodic and Piecewise Continuous Input Functions 4.6 Impulses and Delta Functions   5 Linear Systems of Differential Equations 5.1 First-Order Systems and Applications 5.2 The Method of Elimination 5.3 Matrices and Linear Systems 5.4 The Eigenvalue Method for Homogeneous Systems 5.5 Second-Order Systems and Mechanical Applications 5.6 Multiple Eigenvalue Solutions 5.7 Matrix Exponentials and Linear Systems 5.8 Nonhomogeneous Linear Systems   6 Numerical Methods 6.1 Numerical Approximation: Euler's Method 6.2 A Closer Look at the Euler Method 6.3 The Runge-Kutta Method 6.4 Numerical Methods for Systems   7 Nonlinear Systems and Phenomena 7.1 Equilibrium Solutions and Stability 7.2 Stability and the Phase Plane 7.3 Linear and Almost Linear Systems 7.4 Ecological Models: Predators and Competitors 7.5 Nonlinear Mechanical Systems 7.6 Chaos in Dynamical Systems   8 Eigenvalues and Boundary Value Problems 8.1 Sturm-Liouville Problems and Eigenfunction Expansions 8.2 Applications of Eigenfunction Series 8.3 Steady Periodic Solutions and Natural Frequencies 8.4 Cylindrical Coordinate Problems 8.5 Higher-Dimensional Phenomena   9 Fourier Series Methods 9.1 Periodic Functions and Trigonometric Series 9.2 General Fourier Series and Convergence 9.3 Fourier Sine and Cosine Series 9.4 Applications of Fourier Series 9.5 Heat Conduction and Separation of Variables 9.6 Vibrating Strings and the One-Dimensional Wave Equation 9.7 Steady-State Temperature and Laplace's Equation   Appendix: Existence and Uniqueness of Solutions Answers to Selected Problems Index I-1  

• Proven chapter and section structure of the book is unchanged – Instructors' notes and syllabi will not require revision to continue teaching with the Sixth Edition. • A solid numerical emphasis – Includes generic numerical algorithms and a limited number of illustrative graphic calculator, BASIC, and MATLAB routines. • Wide range of choices regarding breadth and depth of coverage – The first few sections of most chapters introduce the principal ideas of each topic, with remaining sections devoted to extensions and applications. • Computer-generated figures – Includes graphics using Mathematica or MATLAB; shows students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life. • Application Modules – Follow appropriate sections in the book; give students concrete applied emphasis and engages them in more extensive investigations than in typical exercises and problems. • Instructor's Solutions Manual – Provides worked-out solutions for most of the problems in the book. • Student Solutions Manual – Contains solutions for most of the odd-numbered problems.

New graphics and new text inserted where needed for improved student accessibility and understanding of difficult concepts. • Chapter by chapter changes: – Chapter 1: New Figures 1.3.9 and 1.3.10 showing direction fields that illustrate failure of existence and uniqueness of solutions New Problems 34 and 35 showing that small changes in initial conditions can make big differences in results, but big changes in initial conditions may sometimes make only small differences in results; new Remarks 1 and 2 clarifying the concept of implicit solutions; new Remark clarifying the meaning of homogeneity for first-order equations; additional details inserted in the derivation of the rocket propulsion equationand new Problem 5 inserted to investigate the liftoff pause of a rocket on the launch pad sometimes observed before blastoff. – Chapter 2: New explanation of signs and directions of internal forces in mass-spring systems; new introduction of differential operators and clarification of the algebra of polynomial operators; new introduction and illustration of polar exponential forms of complex numbers; fuller explanation of method of undetermined coefficients in Examples 1 and 3; new Remarks 1 and 2 introducing "shooting" terminology, and new Figures 2.8.1 and 2.8.2 illustrating why some endpoint value problems have infinitely many solutions, while others have no solutions at all; new Figures 2.8.4 and 2.8.5 illustrating different types of eigenfunctions. – Chapter 3: New Problem 35 on determination of radii of convergence of power series solutions of differential equations; new Example 3 just before the subsection on logarithmic cases in the method of Frobenius, to illustrate first the reduction-of-order formula with a simple non-series problem. – Chapter 4: New discussion clarifying functions of exponential order and existence of Laplace transforms; new Remark discussing the mechanics of partial-fraction decomposition; new much-expanded discussion of the proof of the Laplace-transform existence theorem and its extension to include the jump discontinuities that play an important role in many practical applications. – Chapter 5: New Problems 20—23 for student exploration of three-railway-cars systems with different initial velocity conditions; new Remark illustrating the relation between matrix exponential methods and the generalized eigenvalue methods discussed previously; new exposition inserted at end of section to explain the connection between matrix variation of parameters here and (scalar) variation of parameters for second-order equations discussed previously in Chapter 3. – Chapter 6: New discussion with new Figures 6.3.11 and 6.3.12 clarifying the difference between rotating and non-rotating coordinate systems in moon-earth orbit problems. – Chapter 7: New remarks on phase plane portraits, autonomous systems, and critical points; new introduction of linearized systems; new 3-dimensional Figures 6.5.18 and 6.5.20 illustrating Lorenz and Rössler trajectories. – Chapter 8: New considerably expanded explanation of even and odd extensions and their Fourier sine-cosine series; new discussion of periodic and non-periodic particular solutions illustrated by new Figure 8.4.4, together with new Problems 19 and 20 at end of section; new example discussion inserted at end of section to illustrate the effects of damping in mass-spring systems; new discussion of signs and direction of heat flow in the derivation of the heat equation. – Chapter 9: Clarification of the effect of internal stretching in deriving the wave equation for longitudinal vibrations of a bar; new Figures 9.5.15