Ordinary Differential Equations and Boundary Value Problems

The authors give a treatment of the theory of ordinary differential equations (ODEs) that is excellent for a first course at the graduate level as well as for individual study. The reader will find it to be a captivating introduction with a number of non-routine exercises dispersed throughout the book.

The authors begin with a study of initial value problems for systems of differential equations including the Picard and Peano existence theorems. The continuability of solutions, their continuous dependence on initial conditions, and their continuous dependence with respect to parameters are presented in detail. This is followed by a discussion of the differentiability of solutions with respect to initial conditions and with respect to parameters. Comparison results and differential inequalities are included as well.

Linear systems of differential equations are treated in detail as is appropriate for a study of ODEs at this level. Just the right amount of basic properties of matrices are introduced to facilitate the observation of matrix systems and especially those with constant coefficients. Floquet theory for linear periodic systems is presented and used to analyze nonhomogeneous linear systems.

Stability theory of first order and vector linear systems are considered. The relationships between stability of solutions, uniform stability, asymptotic stability, uniformly asymptotic stability, and strong stability are examined and illustrated with examples as is the stability of vector linear systems. The book concludes with a chapter on perturbed systems of ODEs.
Contents: Systems of Differential EquationsContinuation of Solutions and Maximal Intervals of ExistenceSmooth Dependence on Initial Conditions and Smooth Dependence on a ParameterSome Comparison Theorems and Differential InequalitiesLinear Systems of Differential EquationsPeriodic Linear Systems and Floquet TheoryStability TheoryPerturbed Systems and More on Existence of Periodic Solutions
Readership: Graduate students and researchers interested in ordinary differential equations.
Keywords:Differential Equations;Linear Systems;Comparison Theorems;Differential Inequalities;Periodic Systems;Floquet Theory;Stability Theory;Perturbed Equations;Periodic SolutionsReview:Key Features:Clarity of presentationTreatment of linear and nonlinear problemsIntroduction to stability theoryNonroutine exercises to expand insight into more difficult conceptsExamples provided with thorough explanations