Differential equations and linear algebra / C. Henry Edwards, David E. Penney.

Includes index.

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 MODULESThe 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 Curves1.4 The Logistic Equation1.5 Indoor Temperature Oscillations1.6 Computer Algebra Solutions2.1 Logistic Modeling of Population Data2.3 Rocket Propulsion2.4 Implementing Euler's Method2.5 Improved Euler Implementation2.6 Runge-Kutta Implementation3.2 Automated Row Operations3.3 Automated Row Reduction3.5 Automated Solution of Linear Systems5.1 Plotting Second-Order Solution Families5.2 Plotting Third-Order Solution Families5.3 Approximate Solutions of Linear Equations5.5 Automated Variation of Parameters5.6 Forced Vibrations and Resonance7.1 Gravitation and Kepler's Laws of Planetary Motion7.3 Automatic Calculation of Eigenvalues and Eigenvectors7.4 Earthquake-Induced Vibrations of Multistory Buildings7.5 Defective Eigenvalues and Generalized Eigenvectors7.6 Comets and Spacecraft8.1 Automated Matrix Exponential Solutions8.2 Automated Variation of Parameters9.1 Phase Portraits and First-Order Equations9.2 Phase Portraits of Almost Linear Systems9.3 Your Own Wildlife Conservation Preserve9.4 The Rayleigh and van der Pol Equations