EXISTENCE THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS

by Francis Joseph Murray, Francis J. Murray, Kenneth S. Miller

Book Overview

This text surveys fundamental and general existence theorems as well as uniqueness theorems and Picard iterants, applying them to properties of solutions and linear differential equations. A basic... This description may be from another edition of this product.

• Format: Unknown Binding
• ISBN: B007YZSA6U
• Release Date: January 1
• Weight: 0.95 lbs.

Customer Reviews

Good Introduction to the Theory of ODE's

Ordinary differential equations are one of the most important topics in applied mathematics. As such there is nothing in this book that is not key. This book manages to be both a brief and reasonably clear introduction to the theory of this field, providing proofs of some the most widely sited and used results in science and engineering as well as other areas of mathematics like differential geometry. It will not help you get better at solving differential equations, but it will help you better understand them and better understand when they have solutions, when those solutions are unique, and what additional properties those solutions might have based on the properties of the differential equations themselves. So for one thing, the title is too narrow for the contents. This book is divided into six chapters, and presents the theory in a logical and progressive order. The first chapter covers a basic existence theorem first in one dependent variable then in multiple dependent variables. The second chapter covers a general existence theorem based on the implicit function theorem which is proven at the outset of this chapter. Chapter three looks at uniqueness and introduces the famous Lipschitz condition in this connection. Chapter four explores Picard iterants and uses the Lipschitz condition to prove convergence of these iterated integrals thus providing an intuitive and useful theoretical tool for exploring questions of the continuity of solutions with respect to initial conditions and/or parameters, and these are the topics that round out chapter four. Chapter five studies additional properties of solutions beyond the ones treated thus far and some sufficient conditions for them. This chapter has by far the fussiest limit arguments in the book and it is also in this chapter that complex analysis first rears its head in connection with the analycity of solutions. Chatper six concludes the book with a look at linear equations, the Jordan normal form of a linear transformation (defined and explained, but not proven), Green functions for ODE's, the monodromy group, and a final section on linear equations with constant coefficients. This book is not without its faults. The Lipschitz condition is defined only for convex domains, and domains are generally assumed to be convex thereafter. But the importance of requiring convexity is never discussed. Ever. This is quite puzzling as most results are local results where convexity is guaranteed anyway. In chapter five, where the going gets tougher, the typo density increases, although there are relatively few typos throughout. In chapter six, the notation is awkward and the author proves several results using division. Division? In a vector space? The author apparently doesn't even think this is worthy of comment, but for me it was a serious breach of mathematical good taste. Finally throughout the book there is very little effort made to visually separate the theorems from the main text, making