Overview

This book is different than most math books because it is based on a student's class notes that have been reviewed and updated by the professor who taught the course! Therefore, this book is structured and presented in a way that teaches the material with a series of detailed examples, from simple to increasing complexity. Solution techniques are generalized and simple, step-by-step procedures are listed and used to solve a series of problems, from simple to advanced. All topics are presented in a logical way that gives the reader a clear understanding of the important concepts and the relationship between topics. This book should be considered a reference for students taking a course in Differential Equations (DEs). Statements of theorem and important results are included; however, proofs are not included because the primary emphasis is on problems and applications. It is assumed that the reader has previously taken courses in Differential and Integral Calculus. Familiarity with Linear Algebra is also recommended. Throughout the book, short reviews are included where needed. In addition, the appendix includes some supporting material (integration-by-parts (tabular-method), complex numbers, linear independence, and matrices). This book includes an impressive list of topics: FIRST-ORDER DE * First Order Linear DE * Separable DE * Substitution Methods (Bernoulli DE, Homogeneous DE, Special Substitution) * Exact DE With & Without Integrating Factor * Strategies for First-Order DE * Existence and Uniqueness Theorem SECOND-ORDER DE * Second-Order DE with Constant Coefficients (Characteristic Eqns., Existence & Uniqueness, Wronskian Determinant) * Higher-Order DE with Constant Coefficients * Non-Homogeneous DE with Constant Coefficients (Undetermined Coefficients, Variation of Parameters) * Euler's DE * Reduction of Order * Mechanical and Electrical Vibrations (Spring Mass System, Electrical Vibrations) SERIES SOLUTIONS of DE * Power Series (Review) * Series Solution near an Ordinary Point * Types of Singular Points * Series Solution near Regular Singular Point * Bessel's DE LAPLACE TRANSFORM of DE * Improper Integration * Definition of Laplace Transform and Gamma Functions * Step Functions * Impulse Functions * Convolution Integrals SYSTEMS of FIRST-ORDER DE * Systems of First-Order Linear DEs * Eigenvalues and Eigenvectors * Homogeneous and Non-Homogeneous Systems * The Phase Plane * Locally Linear Systems * Competing Species * Predator-Prey Systems The last, and perhaps most interesting, chapter of this book includes phase portraits and discusses the stability of equilibrium solutions for non-linear systems of differential equations. The book concludes with fascinating applications such as competing species and predator-prey systems.

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