Features Offers a unique presentation sharply focused on detail Contains illustrative examples and exercises at the end of each chapter Provides an elaboration of details, intended to stimulate students Though ordinary differential equations is taught as a core course to students in mathematics and applied mathematics, detailed coverage of the topics with sufficient examples is unique. Written by a mathematics professor and intended as a textbook for third- and fourth-year undergraduates, the five chapters of this publication give a precise account of higher order differential equations, power series solutions, special functions, existence and uniqueness of solutions, and systems of linear equations. Relevant motivation for different concepts in each chapter and discussion of theory and problems-without the omission of steps-sets Ordinary Differential Equations: A First Course apart from other texts on ODEs. Full of distinguishing examples and containing exercises at the end of each chapter, this lucid course book will promote self-study among students. HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS Introduction Preliminaries Initial Value Problems Boundary Value Problems Superposition Principle The Wronskian and Its Properties Linear Dependence of Solutions Reduction of Order Method of Variation of Parameters The Method of Variation of Parameters for the Non-Homogeneous Equation of n-th order A Formula for the Wronskian Homogeneous Linear Differential Equations with constant Coefficients n-th Order Homogeneous Differential Equations with Constant Coefficients Examples I Exercises I POWER SERIES SOLUTIONS Introduction The Taylor Series Method Second Order Equations with Ordinary Points Second Order Linear Equations with Regular Singular Points Two Exceptional Cases Gauss Hypergeometric Series The Point at Infinity as a Singular Point Examples II Exercises II FUNCTIONS OF DIFFERENTIAL EQUATIONS Introduction Legendre Functions Legendre Series Expansion Some Properties of Legendre Polynomials Hermite Polynomials Properties of Laguerre Polynomials Properties of Bessel Functions Bessel Series Expansion Examples III Exercises III EXISTENCE AND UNIQUENESS OF SOLUTIONS Introduction Lipschitz Condition and Gronwall inequality Successive Approximations and Picard Theorem Dependence of Solutions on the Initial Conditions Dependence of Solutions on the Functions Continuations of the Solutions Non-Local Existence of Solutions Examples IV Exercises IV SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS Introduction Systems of First Order Equations Matrix Preliminaries Representation of n-th Order Equations as a System Existence and Uniqueness of Solutions of System of Equations Wronskian of Vector Functions The Fundamental Matrix and its Properties Non-Homogeneous Linear Systems Linear Systems with Constant Coefficients Linear Systems with Periodic Coefficients Existence and Uniqueness of Solutions of systems Examples V Exercises V REFERENCES SOLUTIONS TO EXERCISES INDEX Instructors We provide complimentary e-inspection copies of primary textbooks to instructors considering our books for course adoption. Request an e-inspection copy Share this Title AddThis Sharing Buttons Related Titles 1 of 1 Solution of Ordinary Differential Equations by Continuous Groups After detailing a few preliminaries on the n-th order ordinary differential equations such as their solutions, initial and boundary value problems, we shall introduce the notion of linear independence and dependence of a set of functions defined on an interval I = [a, b] of the real numbers (or real line) R.
