An Introduction to Ordinary Differential Equations 1st Edition

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This textbook provides a rigorous and lucid introduction to the theory of ordinary differential equations (ODEs), which serve as mathematical models for many exciting real-world problems in science, engineering, and other disciplines. Salient Features Effectively organizes the subject into easily manageable sections in the form of 42 class-tested lectures. Provides a theoretical treatment by organizing the material around theorems and proofs. Uses detailed examples to drive the presentation. Includes numerous exercise sets that encourage pursuing extensions of the material, each with an 'answers or hints' section. Covers an array of advanced topics which allow for flexibility in developing the subject beyond the basics. Provides excellent grounding and inspiration for future research contributions to the field of ODEs and related areas. This book is ideal for a senior undergraduate or a graduate-level course on ordinary differential equations. Prerequisites include a course in calculus.

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Preface Introduction Historical Notes Exact Equations Elementary First Order Equations First Order Linear Equations Second Order Linear Equations Preliminaries to Existence and Uniqueness of Solutions Picard’s Method of Successive Approximations Existence Theorems Uniqueness Theorems Differential Inequalities Continuous Dependence on Initial Conditions Preliminary Results from Algebra and Analysis Preliminary Results from Algebra and Analysis (Contd.) Existence and Uniqueness of Solutions of Systems Existence and Uniqueness of Solutions of Systems (Contd.) General Properties of Linear Systems Fundamental Matrix Solution Systems with Constant Coefficients Periodic Linear Systems Asymptotic Behavior of Solutions of Linear Systems Asymptotic Behavior of Solutions of Linear Systems (Contd.) Preliminaries to Stability of Solutions Stability of Quasi–Linear Systems Two–Dimensional Autonomous Systems Two–Dimensional Autonomous Systems (Contd.) Limit Cycles and Periodic Solutions Lyapunov’s Direct Method for Autonomous Systems Lyapunov’s Direct Method for Non–Autonomous Systems Higher Order Exact and Adjoint Equations.- Oscillatory Equations Linear Boundary Value Problems Green’s Functions Degenerate Linear Boundary Value Problems Maximum Principles Sturm–Liouville Problems Sturm–Liouville Problems (Contd.) Eigenfunction Expansions Eigenfunction Expansions (Contd.) Nonlinear Boundary Value Problems Nonlinear Boundary Value Problems (Contd.) Topics for Further Studies References Index