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Existence Theorems for Ordinary Differential Equations (Dover Books on Mathematics) Paperback – June 5, 2007


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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 176 pages
  • Publisher: Dover Publications; Dover Ed edition (June 5, 2007)
  • Language: English
  • ISBN-10: 0486458105
  • ISBN-13: 978-0486458106
  • Product Dimensions: 0.5 x 5.5 x 8.5 inches
  • Shipping Weight: 7 ounces (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #3,211,120 in Books (See Top 100 in Books)

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7 of 7 people found the following review helpful By Jason Dowd on February 5, 2010
Format: Paperback Verified Purchase
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.
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