Differential Equations and Linear Algebra -- ACCUMULATOR

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.  David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

• Table of Contents First-Order Differential Equations 1.1 Differential Equations and Mathematical Models1.2 Integrals as General and Particular Solutions1.3 Slope Fields and Solution Curves1.4 Separable Equations and Applications1.5 Linear First-Order Equations1.6 Substitution Methods and Exact Equations Mathematical Models and Numerical Methods 2.1 Population Models2.2 Equilibrium Solutions and Stability2.3 Acceleration–Velocity Models2.4 Numerical Approximation: Euler's Method2.5 A Closer Look at the Euler Method2.6 The Runge–Kutta Method Linear Systems and Matrices 3.1 Introduction to Linear Systems3.2 Matrices and Gaussian Elimination3.3 Reduced Row-Echelon Matrices3.4 Matrix Operations3.5 Inverses of Matrices3.6 Determinants3.7 Linear Equations and Curve Fitting Vector Spaces 4.1 The Vector Space R3 4.2 The Vector Space Rn and Subspaces4.3 Linear Combinations and Independence of Vectors4.4 Bases and Dimension for Vector Spaces4.5 Row and Column Spaces4.6 Orthogonal Vectors in Rn4.7 General Vector Spaces Higher-Order Linear Differential Equations 5.1 Introduction: Second-Order Linear Equations5.2 General Solutions of Linear Equations5.3 Homogeneous Equations with Constant Coefficients5.4 Mechanical Vibrations5.5 Nonhomogeneous Equations and Undetermined Coefficients5.6 Forced Oscillations and Resonance Eigenvalues and Eigenvectors 6.1 Introduction to Eigenvalues6.2 Diagonalization of Matrices6.3 Applications Involving Powers of Matrices Linear Systems of Differential Equations 7.1 First-Order Systems and Applications7.2 Matrices and Linear Systems7.3 The Eigenvalue Method for Linear Systems7.4 A Gallery of Solution Curves of Linear Systems7.5 Second-Order Systems and Mechanical Applications7.6 Multiple Eigenvalue Solutions7.7 Numerical Methods for Systems Matrix Exponential Methods 8.1 Matrix Exponentials and Linear Systems8.2 Nonhomogeneous Linear Systems8.3 Spectral Decomposition Methods Nonlinear Systems and Phenomena 9.1 Stability and the Phase Plane9.2 Linear and Almost Linear Systems9.3 Ecological Models: Predators and Competitors9.4 Nonlinear Mechanical Systems Laplace Transform Methods 10.1 Laplace Transforms and Inverse Transforms10.2 Transformation of Initial Value Problems10.3 Translation and Partial Fractions10.4 Derivatives, Integrals, and Products of Transforms10.5 Periodic and Piecewise Continuous Input Functions Power Series Methods 11.1 Introduction and Review of Power Series11.2 Power Series Solutions11.3 Frobenius Series Solutions11.4 Bessel FunctionsAppendix A: Existence and Uniqueness of Solutions Appendix B: Theory of Determinants APPLICATION MODULES The modules listed below follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Many of these modules are enhanced by the supplementary material found at the new Expanded Applications website. 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 Dynamic Phase Plane Graphics7.5 Earthquake-Induced Vibrations of Multistory Buildings7.6 Defective Eigenvalues and Generalized Eigenvectors7.7 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