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Über dieses Buch

This textbook is designed with the needs of today’s student in mind. It is the ideal textbook for a first course in elementary differential equations for future engineers and scientists, including mathematicians. This book is accessible to anyone who has a basic knowledge of precalculus algebra and differential and integral calculus. Its carefully crafted text adopts a concise, simple, no-frills approach to differential equations, which helps students acquire a solid experience in many classical solution techniques. With a lighter accent on the physical interpretation of the results, a more manageable page count than comparable texts, a highly readable style, and over 1000 exercises designed to be solved without a calculating device, this book emphasizes the understanding and practice of essential topics in a succinct yet fully rigorous fashion. Apart from several other enhancements, the second edition contains one new chapter on numerical methods of solution.

The book formally splits the "pure" and "applied" parts of the contents by placing the discussion of selected mathematical models in separate chapters. At the end of most of the 246 worked examples, the author provides the commands in Mathematica® for verifying the results. The book can be used independently by the average student to learn the fundamentals of the subject, while those interested in pursuing more advanced material can regard it as an easily taken first step on the way to the next level. Additionally, practitioners who encounter differential equations in their professional work will find this text to be a convenient source of reference.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
Mathematical modeling is one of the most important and powerful methods for studying phenomena occurring in our universe. Generally speaking, such a model is made up of one or several equations from which we aim to determine one or several unknown quantities of interest in terms of other, prescribed, quantities. The unknown quantities turn out in many cases to be functions of a set of variables.
Christian Constanda

Chapter 2. First-Order Equations

Abstract
Certain types of first-order equations can be solved by relatively simple methods. Since, as seen in Sect. 1.​2, many mathematical models are constructed with such equations, it is important to get familiarized with their solution procedures.
Christian Constanda

Chapter 3. Mathematical Models with First-Order Equations

Abstract
In Sect. 1.​2 we listed examples of DEs arising in some mathematical models. We now show how these equations are derived, and find their solutions under suitable ICs.
Christian Constanda

Chapter 4. Linear Second-Order Equations

Abstract
A large number of mathematical models, particularly in the physical sciences and engineering, consist of IVPs or BVPs for second-order DEs.
Christian Constanda

Chapter 5. Mathematical Models with Second-Order Equations

Abstract
In this chapter, we illustrate the use of linear second-order equations with constant coefficients in the analysis of mechanical oscillations and electrical vibrations.
Christian Constanda

Chapter 6. Higher-Order Linear Equations

Abstract
Certain physical phenomena give rise to mathematical models that involve DEs of an order higher than two.
Christian Constanda

Chapter 7. Systems of Differential Equations

Abstract
As physical phenomena increase in complexity, their mathematical models require the use of more than one unknown function. This gives rise to systems of DEs.
Christian Constanda

Chapter 8. The Laplace Transformation

Abstract
The purpose of an analytic transformation is to change a more complicated problem into a simpler one. The Laplace transformation, which is applied chiefly with respect to the time variable, maps an IVP onto an algebraic equation or system.
Christian Constanda

Chapter 9. Series Solutions

Abstract
Owing to the complicated structure of some DEs, it is not always possible to obtain the exact solution of an IVP. In such situations, we need to resort to methods that produce an approximate solution, which is usually constructed in the form of an infinite series. In what follows we illustrate a procedure of this type, based on series expansions for functions of a real variable.
Christian Constanda

Chapter 10. Numerical Methods

Abstract
In Chap. 2, we considered a few types of first-order differential equations for which we found exact solutions. Unfortunately, the great majority of DEs arising in mathematical models cannot be solved explicitly, so we have to apply other techniques in order to obtain details about the nature and properties of their solutions. A direction field sketch, for example (see Sect. 2.8), helps us understand the long term behavior of solution families. But if we require quantitative information about their evolution on finite intervals, we need to resort to numerical approximation techniques. In what follows, we introduce four such methods that appeal to the user because of their combination of simplicity and, in many cases, reasonable accuracy. All numerical results are computed with rounding at the fourth decimal place.
Christian Constanda

Backmatter

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