Differential Equations and Linear Algebra

For courses in Differential Equations and Linear Algebra. Acclaimed authors Edwards and Penney combine core topics in elementary differential equations with those concepts and methods of elementary linear algebra needed for a contemporary combined introduction to differential equations and linear algebra. Known for its real-world applications and its blend of algebraic and geometric approaches, this text discusses mathematical modeling of real-world phenomena, with a fresh new computational and qualitative flavor evident throughout in figures, examples, problems, and applications. In the Third Edition, new graphics and narrative have been added as needed-yet the proven chapter and section structure remains unchanged, so that class notes and syllabi will not require revision for the new edition.CHAPTER 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 CHAPTER 2. Mathematical Models and Numerical Methods 2.1 Population Models 2.2 Equilibrium Solutions and Stability 2.3 Acceleration-Velocity Models 2.4 Numerical Approximation: Euler's Method 2.5 A Closer Look at the Euler Method 2.6 The Runge-Kutta Method CHAPTER 3. Linear Systems and Matrices 3.1 Introduction to Linear Systems 3.2 Matrices and Gaussian Elimination 3.3 Reduced Row-Echelon Matrices 3.4 Matrix Operations 3.5 Inverses of Matrices 3.6 Determinants 3.7 Linear Equations and Curve Fitting CHAPTER 4. Vector Spaces 4.1 The Vector Space R3 4.2 The Vector Space Rn and Subspaces 4.3 Linear Combinations and Independence of Vectors 4.4 Bases and Dimension for Vector Spaces 4.5 Row and Column Spaces 4.6 Orthogonal Vectors in Rn 4.7 General Vector Spaces CHAPTER 5. Higher-Order Linear Differential Equations 5.1 Introduction: Second-Order Linear Equations 5.2 General Solutions of Linear Equations 5.3 Homogeneous Equations with Constant Coefficients 5.4 Mechanical Vibrations 5.5 Nonhomogeneous Equations and Undetermined Coefficients 5.6 Forced Oscillations and Resonance CHAPTER 6. Eigenvalues and Eigenvectors 6.1 Introduction to Eigenvalues 6.2 Diagonalization of Matrices 6.3 Applications Involving Powers of Matrices CHAPTER 7. Linear Systems of Differential Equations 7.1 First-Order Systems and Applications 7.2 Matrices and Linear Systems 7.3 The Eigenvalue Method for Linear Systems 7.4 Second-Order Systems and Mechanical Applications 7.5 Multiple Eigenvalue Solutions 7.6 Numerical Methods for Systems CHAPTER 8. Matrix Exponential Methods 8.1 Matrix Exponentials and Linear Systems 8.2 Nonhomogeneous Linear Systems 8.3 Spectral Decomposition Methods CHAPTER 9. Nonlinear Systems and Phenomena 9.1 Stability and the Phase Plane 9.2 Linear and Almost Linear Systems 9.3 Ecological Models: Predators and Competitors 9.4 Nonlinear Mechanical Systems CHAPTER 10. Laplace Transform Methods 10.1 Laplace Transforms and Inverse Transforms 10.2 Transformation of Initial Value Problems 10.3 Translation and Partial Fractions 10.4 Derivatives, Integrals, and Products of Transforms 10.5 Periodic and Piecewise Continuous Input Functions CHAPTER 11. Power Series Methods 11.1 Introduction and Review of Power Series 11.2 Power Series Solutions 11.3 Frobenius Series Solutions 11.4 Bessel Functions References for Further Study Appendix A: Existence and Uniqueness of Solutions Appendix B: Theory of Determinants Answers to Selected Problems Index APPLICATION MODULES The modules listed here follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook. 1.3 Computer-Generated Slope Fields and Solution Curves 1.4 The Logistic Equation 1.5 Indoor Temperature Oscillations 1.6 Computer Algebra Solutions 2.1 Logistic Modeling of Population Data 2.3 Rocket Propulsion 2.4 Implementing Euler's Method 2.5 Improved Euler Implementation 2.6 Runge-Kutta Implementation 3.2 Automated Row Operations 3.3 Automated Row Reduction 3.5 Automated Solution of Linear Systems 5.1 Plotting Second-Order Solution Families 5.2 Plotting Third-Order Solution Families 5.3 Approximate Solutions of Linear Equations 5.5 Automated Variation of Parameters 5.6 Forced Vibrations and Resonance 7.1 Gravitation and Kepler's Laws of Planetary Motion 7.3 Automatic Calculation of Eigenvalues and Eigenvectors 7.4 Earthquake-Induced Vibrations of Multistory Buildings 7.5 Defective Eigenvalues and Generalized Eigenvectors 7.6 Comets and Spacecraft 8.1 Automated Matrix Exponential Solutions 8.2 Automated Variation of Parameters 9.1 Phase Portraits and First-Order Equations 9.2 Phase Portraits of Almost Linear Systems 9.3 Your Own Wildlife Conservation Preserve 9.4 The Rayleigh and van der Pol Equations