Fundamentals of Differential Equations

R. Kent Nagle (deceased) taught at the University of South Florida. He was a research mathematician and an accomplished author. His legacy is honored in part by the Nagle Lecture Series which promotes mathematics education and the impact of mathematics on society. He was a member of the American Mathematical Society for 21 years. Throughout his life, he imparted his love for mathematics to everyone, from students to colleagues.  Edward B. Saff received his B.S. in applied mathematics from Georgia Institute of Technology and his Ph.D. in Mathematics from the University of Maryland. After his tenure as Distinguished Research Professor at the University of South Florida, he joined the Vanderbilt University Mathematics Department faculty in 2001 as Professor and Director of the Center for Constructive Approximation. His research areas include approximation theory, numerical analysis, and potential theory. He has published more than 240 mathematical research articles, co-authored 9 books, and co-edited 11 volumes. Other recognitions of his research include his election as a Foreign Member of the Bulgarian Academy of Sciences (2013); and as a Fellow of the American Mathematical Society (2013). He is particularly active on the international scene, serving as an advisor and NATO collaborator to a French research team at INRIA Sophia-Antipolis; a co-director of an Australian Research Council Discovery Award; an annual visiting research collaborator at the University of Cyprus in Nicosia; and as an organizer of a sequence of international research conferences that helps foster the careers of mathematicians from developing countries.  Arthur David Snider has 50+ years of experience in modeling physical systems in the areas of heat transfer, electromagnetics, microwave circuits, and orbital mechanics, as well as the mathematical areas of numerical analysis, signal processing, differential equations, and optimization. He holds degrees in mathematics (BS, MIT; PhD, NYU) and physics (MA, Boston U), and is a registered professional engineer. He served 45 years on the faculties of mathematics, physics, and electrical engineering at the University of South Florida. He worked 5 years as a systems analyst at MIT's Draper Instrumentation Lab, and has consulted for General Electric, Honeywell, Raytheon, Texas, Instruments, Kollsman, E-Systems, Harris, and Intersil. He has authored nine textbooks and roughly 100 journal articles. Hobbies include bluegrass fiddle, acting, and handball.

• 1. Introduction 1.1 Background1.2 Solutions and Initial Value Problems1.3 Direction Fields1.4 The Approximation Method of Euler2. First-Order Differential Equations 2.1 Introduction: Motion of a Falling Body2.2 Separable Equations2.3 Linear Equations2.4 Exact Equations2.5 Special Integrating Factors2.6 Substitutions and Transformations3. Mathematical Models and Numerical Methods Involving First Order Equations 3.1 Mathematical Modeling3.2 Compartmental Analysis3.3 Heating and Cooling of Buildings3.4 Newtonian Mechanics3.5 Electrical Circuits3.6 Improved Euler's Method3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta4. Linear Second-Order Equations 4.1 Introduction: The Mass-Spring Oscillator4.2 Homogeneous Linear Equations: The General Solution4.3 Auxiliary Equations with Complex Roots4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients4.5 The Superposition Principle and Undetermined Coefficients Revisited4.6 Variation of Parameters4.7 Variable-Coefficient Equations4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations4.9 A Closer Look at Free Mechanical Vibrations4.10 A Closer Look at Forced Mechanical Vibrations5. Introduction to Systems and Phase Plane Analysis 5.1 Interconnected Fluid Tanks5.2 Elimination Method for Systems with Constant Coefficients5.3 Solving Systems and Higher-Order Equations Numerically5.4 Introduction to the Phase Plane5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models5.6 Coupled Mass-Spring Systems5.7 Electrical Systems5.8 Dynamical Systems, Poincaré Maps, and Chaos6. Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory of Linear Differential Equations6.2 Homogeneous Linear Equations with Constant Coefficients6.3 Undetermined Coefficients and the Annihilator Method6.4 Method of Variation of Parameters7. Laplace Transforms 7.1 Introduction: A Mixing Problem7.2 Definition of the Laplace Transform7.3 Properties of the Laplace Transform7.4 Inverse Laplace Transform7.5 Solving Initial Value Problems7.6 Transforms of Discontinuous Functions7.7 Transforms of Periodic and Power Functions7.8 Convolution7.9 Impulses and the Dirac Delta Function7.10 Solving Linear Systems with Laplace Transforms8. Series Solutions of Differential Equations 8.1 Introduction: The Taylor Polynomial Approximation8.2 Power Series and Analytic Functions8.3 Power Series Solutions to Linear Differential Equations8.4 Equations with Analytic Coefficients8.5 Cauchy-Euler (Equidimensional) Equations8.6 Method of Frobenius8.7 Finding a Second Linearly Independent Solution8.8 Special Functions9. Matrix Methods for Linear Systems 9.1 Introduction9.2 Review 1: Linear Algebraic Equations9.3 Review 2: Matrices and Vectors9.4 Linear Systems in Normal Form9.5 Homogeneous Linear Systems with Constant Coefficients9.6 Complex Eigenvalues9.7 Nonhomogeneous Linear Systems9.8 The Matrix Exponential Function10. Partial Differential Equations 10.1 Introduction: A Model for Heat Flow10.2 Method of Separation of Variables10.3 Fourier Series10.4 Fourier Cosine and Sine Series10.5 The Heat Equation10.6 The Wave Equation10.7 Laplace's EquationAppendices Newton’s Method Simpson’s RuleCramer’s RuleMethod of Least SquaresRunge-Kutta Procedure for n Equations