Differential Equations and Linear Algebra, 4e Henry Edwards, David Penney, David Calvis (eBook PDF)

Differential Equations & Linear Algebra Contents Application Modules Preface Principal Features of This Revision Features of This Text Supplements 1 First-Order Differential Equations 1.1 Differential Equations a ...

Differential Equations & Linear Algebra Contents Application Modules Preface Principal Features of This Revision Features of This Text Supplements 1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models Differential Equations and Mathematical Models Mathematical Models Examples and Terminology Solution Solution 1.1 Problems Differential Equations as Models 1.2 Integrals as General and Particular Solutions Solution Velocity and Acceleration Solution Physical Units Vertical Motion with Gravitational Acceleration A Swimmer’s Problem 1.2 Problems Velocity Given Graphically 1.3 Slope Fields and Solution Curves Slope Fields and Graphical Solutions Solution Applications of Slope Fields Existence and Uniqueness of Solutions 1.3 Problems 1.3 Application Computer-Generated Slope Fields and Solution Curves 1.4 Separable Equations and Applications Solution Implicit, General, and Singular Solutions Solution Natural Growth and Decay The Natural Growth Equation Solution Solution Cooling and Heating Solution Torricelli’s Law Solution 1.4 Problems Torricelli’s Law 1.4 Application The Logistic Equation 1.5 Linear First-Order Equations Solution Solution A Closer Look at the Method Solution Mixture Problems Solution Solution 1.5 Problems Mixture Problems Polluted Reservoir 1.5 Application Indoor Temperature Oscillations 1.6 Substitution Methods and Exact Equations Solution Homogeneous Equations Solution Solution Bernoulli Equations Flight Trajectories Solution Exact Differential Equations Solution Reducible Second-Order Equations Solution Solution 1.6 Problems 1.6 Application Computer Algebra Solutions Chapter 1 Summary Chapter 1 Review Problems 2 Mathematical Models and Numerical Methods 2.1 Population Models Bounded Populations and the Logistic Equation Limiting Populations and Carrying Capacity Solution Historical Note More Applications of the Logistic Equation Solution Doomsday versus Extinction Solution 2.1 Problems 2.1 Application Logistic Modeling of Population Data 2.2 Equilibrium Solutions and Stability Stability of Critical Points Harvesting a Logistic Population Bifurcation and Dependence on Parameters 2.2 Problems Constant-Rate Harvesting 2.3 Acceleration–Velocity Models Resistance Proportional to Velocity Solution Resistance Proportional to Square of Velocity Variable Gravitational Acceleration Solution Escape Velocity 2.3 Problems 2.3 Application Rocket Propulsion Constant Thrust No Resistance Free Space 2.4 Numerical Approximation: Euler’s Method Solution Local and Cumulative Errors A Word of Caution Solution 2.4 Problems 2.4 Application Implementing Euler’s Method Famous Numbers Investigation 2.5 A Closer Look at the Euler Method Solution An Improvement in Euler’s Method Answer 2.5 Problems Decreasing Step Size 2.5 Application Improved Euler Implementation Famous Numbers Revisited Logistic Population Investigation Periodic Harvesting and Restocking 2.6 The Runge–Kutta Method Solution 2.6 Problems Velocity-Acceleration Problems 2.6 Application Runge–Kutta Implementation Famous Numbers Revisited, One Last Time The Skydiver’s Descent 3 Linear Systems and Matrices 3.1 Introduction to Linear Systems Two Equations in Two Unknowns Three Possibilities The Method of Elimination Three Equations in Three Unknowns Solution Solution A Differential Equations Application 3.1 Problems 3.2 Matrices and Gaussian Elimination Coefficient Matrices Elementary Row Operations Echelon Matrices Gaussian Elimination 3.2 Problems 3.2 Application Automated Row Reduction 3.3 Reduced Row-Echelon Matrices Solution The Three Possibilities Homogeneous Systems Equal Numbers of Equations and Variables 3.3 Problems 3.3 Application Automated Row Reduction 3.4 Matrix Operations Vectors Matrix Multiplication Matrix Equations Matrix Algebra 3.4 Problems 3.5 Inverses of Matrices The Inverse Matrix A-1 How to Find A-1 Solution Matrix Equations Solution Nonsingular Matrices 3.5 Problems 3.5 Application Automated Solution of Linear Systems 3.6 Determinants Higher-Order Determinants Row and Column Properties The Transpose of a Matrix Determinants and Invertibility Cramer’s Rule for n×n Systems Solution Inverses and the Adjoint Matrix Solution Computational Efficiency 3.6 Problems 3.7 Linear Equations and Curve Fitting Solution Modeling World Population Growth Geometric Applications Solution Solution 3.7 Problems Population Modeling 4 Vector Spaces 4.1 The Vector Space R3 The Vector Space R2 Solution Linear Independence in R3 Basis Vectors in R3 Subspaces of R3 4.1 Problems 4.2 The Vector Space Rn and Subspaces Definition of a Vector Space Subspaces 4.2 Problems 4.3 Linear Combinations and Independence of Vectors Linear Independence 4.3 Problems 4.4 Bases and Dimension for Vector Spaces Bases for Solution Spaces Solution 4.4 Problems 4.5 Row and Column Spaces Row Space and Row Rank Column Space and Column Rank Solution Rank and Dimension Nonhomogeneous Linear Systems 4.5 Problems 4.6 Orthogonal Vectors in Rn Solution Orthogonal Complements 4.6 Problems 4.7 General Vector Spaces Function Spaces Solution Solution Solution Spaces of Differential Equations 4.7 Problems 5 Higher-Order Linear Differential Equations 5.1 Introduction: Second-Order Linear Equations A Typical Application Homogeneous Second-Order Linear Equations Solution Linearly Independent Solutions General Solutions Linear Second-Order Equations with Constant Coefficients Solution 5.1 Problems 5.1 Application Plotting Second-Order Solution Families 5.2 General Solutions of Linear Equations Existence and Uniqueness of Solutions Linearly Independent Solutions Solution Solution General Solutions Nonhomogeneous Equations Solution 5.2 Problems 5.2 Application Plotting Third-Order Solution Families 5.3 Homogeneous Equations with Constant Coefficients The Characteristic Equation Distinct Real Roots Solution Polynomial Differential Operators Repeated Real Roots Solution Complex-Valued Functions and Euler’s Formula Complex Roots Solution Solution Repeated Complex Roots Solution Solution 5.3 Problems 5.3 Application Approximate Solutions of Linear Equations 5.4 Mechanical Vibrations The Simple Pendulum Free Undamped Motion Solution Free Damped Motion Solution 5.4 Problems Simple Pendulum Free Damped Motion Differential Equations and Determinism 5.5 Nonhomogeneous Equations and Undetermined Coefficients Solution Solution Solution Solution The General Approach Solution Solution Solution The Case of Duplication Solution Solution Solution Variation of Parameters Solution 5.5 Problems 5.5 Application Automated Variation of Parameters 5.6 Forced Oscillations and Resonance Undamped Forced Oscillations Solution Beats Resonance Modeling Mechanical Systems Solution Solution Damped Forced Oscillations Solution 5.6 Problems Automobile Vibrations 5.6 Application Forced Vibrations 6 Eigenvalues and Eigenvectors 6.1 Introduction to Eigenvalues The Characteristic Equation Solution Solution Eigenspaces Solution 6.1 Problems 6.2 Diagonalization of Matrices Similarity and Diagonalization 6.2 Problems 6.3 Applications Involving Powers of Matrices Solution Transition Matrices Predator-Prey Models The Cayley-Hamilton Theorem 6.3 Problems Predator-Prey 7 Linear Systems of Differential Equations 7.1 First-Order Systems and Applications Initial Applications First-Order Systems Solution Simple Two-Dimensional Systems Linear Systems 7.1 Problems 7.1 Application Gravitation and Kepler’s Laws of Planetary Motion 7.2 Matrices and Linear Systems First-Order Linear Systems Independence and General Solutions Initial Value Problems and Elementary Row Operations Solution Nonhomogeneous Solutions 7.2 Problems 7.3 The Eigenvalue Method for Linear Systems The Eigenvalue Method Distinct Real Eigenvalues Solution Compartmental Analysis Solution Complex Eigenvalues Solution Solution 7.3 Problems Cascading Brine Tanks Interconnected Brine Tanks Open Three-Tank System Closed Three-Tank System 7.3 Application Automatic Calculation of Eigenvalues and Eigenvectors 7.4 A Gallery of Solution Curves of Linear Systems Systems of Dimension n=2 Real Eigenvalues Saddle Points Nodes: Sinks and Sources Zero Eigenvalues and Straight-Line Solutions Repeated Eigenvalues; Proper and Improper Nodes The Special Case of a Repeated Zero Eigenvalue Complex Conjugate Eigenvalues and Eigenvectors Pure Imaginary Eigenvalues: Centers and Elliptical Orbits Solution Complex Eigenvalues: Spiral Sinks and Sources Solution Solution A 3-Dimensional Example 7.4 Problems 7.4 Application Dynamic Phase Plane Graphics 7.5 Second-Order Systems and Mechanical Applications* Solution of Second-Order Systems Forced Oscillations and Resonance Periodic and Transient Solutions 7.5 Problems The Two-Axle Automobile 7.5 Application Earthquake-Induced Vibrations of Multistory Buildings 7.6 Multiple Eigenvalue Solutions Solution Defective Eigenvalues The Case of Multiplicity k=2 Solution Generalized Eigenvectors Solution The General Case An Application The Jordan Normal Form The General Cayley-Hamilton Theorem 7.6 Problems 7.6 Application Defective Eigenvalues and Generalized Eigenvectors 7.7 Numerical Methods for Systems Euler Methods for Systems The Runge–Kutta Method and Second-Order Equations Higher-Order Systems Solution Variable Step Size Methods Earth–Moon Satellite Orbits 7.7 Problems Batted Baseball 7.7 Application Comets and Spacecraft Your Spacecraft Landing Kepler’s Law of Planetary (or Satellite) Motion Halley’s Comet Your Own Comet 8 Matrix Exponential Methods 8.1 Matrix Exponentials and Linear Systems Fundamental Matrix Solutions Solution Exponential Matrices Matrix Exponential Solutions Solution General Matrix Exponentials Solution 8.1 Problems 8.1 Application Automated Matrix Exponential Solutions 8.2 Nonhomogeneous Linear Systems Undetermined Coefficients Solution Variation of Parameters Solution 8.2 Problems Two Brine Tanks 8.2 Application Automated Variation of Parameters 8.3 Spectral Decomposition Methods The Case of Distinct Eigenvalues Second-Order Linear Systems The General Case 8.3 Problems 9 Nonlinear Systems and Phenomena 9.1 Stability and the Phase Plane Solution Phase Portraits Critical Point Behavior Stability Asymptotic Stability 9.1 Problems 9.1 Application Phase Plane Portraits and First-Order Equations 9.2 Linear and Almost Linear Systems Linearization Near a Critical Point Isolated Critical Points of Linear Systems Almost Linear Systems Solution Solution 9.2 Problems Bifurcations 9.2 Application Phase Plane Portraits of Almost Linear Systems 9.3 Ecological Models: Predators and Competitors Competing Species Interactions of Logistic Populations 9.3 Problems Predator–Prey System Competition System Competition System Logistic Prey Population Doomsday vs. Extinction 9.3 Application Your Own Wildlife Conservation Preserve 9.4 Nonlinear Mechanical Systems The Position–Velocity Phase Plane Damped Nonlinear Vibrations The Nonlinear Pendulum Period of Undamped Oscillation Damped Pendulum Oscillations 9.4 Problems Critical Points for Damped Pendulum Critical Points for Mass-Spring System Critical Points for Physical Systems Period of Oscillation 9.4 Application The Rayleigh, van der Pol, and FitzHugh-Nagumo Equations Rayleigh’s Equation Van der Pol’s Equation The FitzHugh-Nagumo Equations 10 Laplace Transform Methods 10.1 Laplace Transforms and Inverse Transforms Linearity of Transforms Inverse Transforms Piecewise Continuous Functions Solution General Properties of Transforms 10.1 Problems 10.1 Application Computer Algebra Transforms and Inverse Transforms 10.2 Transformation of Initial Value Problems Solution of Initial Value Problems Solution Solution Linear Systems Solution The Transform Perspective Additional Transform Techniques Solution Solution Solution Extension of Theorem 1 10.2 Problems 10.2 Application Transforms of Initial Value Problems 10.3 Translation and Partial Fractions Solution Solution Solution Solution Resonance and Repeated Quadratic Factors Solution Solution 10.3 Problems 10.3 Application Damping and Resonance Investigations 10.4 Derivatives, Integrals, and Products of Transforms Differentiation of Transforms Solution Solution Integration of Transforms Solution Solution * Proofs of Theorems 10.4 Problems 10.5 Periodic and Piecewise Continuous Input Functions Solution Solution Solution Transforms of Periodic Functions Solution 10.5 Problems 10.5 Application Engineering Functions 11 Power Series Methods 11.1 Introduction and Review of Power Series Power Series Operations The Power Series Method Solution Shift of Index of Summation Solution Solution Solution 11.1 Problems 11.2 Power Series Solutions Solution Solution Translated Series Solutions Solution Types of Recurrence Relation Solution The Legendre Equation 11.2 Problems 11.2 Application Automatic Computation of Series Coefficients 11.3 Frobenius Series Solutions Types of Singular Points The Method of Frobenius Solution Frobenius Series Solutions Solution Solution When r1-r2 Is an Integer Solution Summary 11.3 Problems 11.3 Application Automating the Frobenius Series Method 11.4 Bessel Functions The Case r=p > 0 The Case r = -p < 0 The Gamma Function Bessel Functions of the First Kind Bessel Functions of the Second Kind Bessel Function Identities Applications of Bessel Functions Solution Solution 11.4 Problems References for Further Study APPENDIX A Existence and Uniqueness of Solutions A.1 Existence of Solutions A.2 Linear Systems A.3 Local Existence A.4 Uniqueness of Solutions A.5 Well-Posed Problems and Mathematical Models Problems APPENDIX B Theory of Determinants Determinants and Elementary Row Operations Determinants and Invertibility Cramer’s Rule and Inverse Matrices Inverses and the Adjoint Matrix Answers to Selected Problems Chapter 1 Section 1.1 Section 1.2 Section 1.3 Section 1.4 Section 1.5 Section 1.6 Chapter 1 Review Problems Chapter 2 Section 2.1 Section 2.2 Section 2.3 Section 2.4 Section 2.5 Section 2.6 Chapter 3 Section 3.1 Section 3.2 Section 3.3 Section 3.4 Section 3.5 Section 3.6 Section 3.7 Chapter 4 Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6 Section 4.7 Chapter 5 Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 Section 5.6 Chapter 6 Section 6.1 Section 6.2 Section 6.3 Chapter 7 Section 7.1 Section 7.2 Section 7.3 Section 7.4 Section 7.5 Section 7.6 Section 7.7 Chapter 8 Section 8.1 Section 8.2 Section 8.3 Chapter 9 Section 9.1 Section 9.2 Section 9.3 Section 9.4 Chapter 10 Section 10.1 Section 10.2 Section 10.3 Section 10.4 Section 10.5 Chapter 11 Section 11.1 Section 11.2 Section 11.3 Section 11.4 Appendix A