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2013 | Book

Introduction Read chapter Read first chapter

Differential Equations

A Primer for Scientists and Engineers

Author: Christian Constanda

Publisher: Springer New York

Book Series : Springer Undergraduate Texts in Mathematics and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Differential Equations for Scientists and Engineers is a book designed with students in mind. It attempts to take a concise, simple, and no-frills approach to differential equations. The approach used in this text is to give students extensive experience in main solution techniques with a lighter emphasis on the physical interpretation of the results. With a more manageable page count than comparable titles, and over 400 exercises that can be solved without a calculating device, this book emphasizes the understanding and practice of essential topics in a succinct fashion. At the end of each worked example, the author provides the Mathematica commands that can be used to check the results and where applicable, to generate graphical representations. It can be used independently by the average student, while those continuing with the subject will develop a fundamental framework with which to pursue more advanced material. This book is designed for undergraduate students with some basic knowledge of precalculus algebra and a first course in calculus.

Table of Contents

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. Since very often the physical or empirical laws governing evolutionary processes implicate the rates of change of these functions with respect to their variables, and since rates of change are represented in mathematics by derivatives, it is important for us to gain knowledge of how to solve equations where the unknown functions occur together with their derivatives.
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. Among the latter, a very important role is played by linear equations. Even if the model is nonlinear, the study of its linearized version can provide valuable hints about the quantitative and qualitative behavior of the full model, and perhaps suggest a method that might lead to its complete solution.
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. Once the latter is solved, its solution is fed into the inverse transformation to yield the solution of the original IVP.
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.
Christian Constanda
Appendix A. Algebra Techniques
Abstract
The integration of a rational function—that is, a function of the form P∕ Q, where P and Q are polynomials—becomes much easier if the function can be written as a sum of simpler expressions, commonly referred to as partial fractions.
Christian Constanda
Appendix B. Calculus Techniques
Abstract
If f = f(x) is a continuous function on an interval J and a and b are two points in J such that f(a) and f(b) have opposite signs, then, by the intermediate value theorem, there is at least one point c between a and b such that f(c) = 0.
Christian Constanda
Appendix C. Table of Laplace Transforms
Christian Constanda
Appendix D. The Greek Alphabet
Abstract
Below is a table of the Greek letters most used by mathematicians. The recommended pronunciation of these letters as symbols in an academic context, listed in the third column, is that of classical, not modern, Greek.
Christian Constanda
Backmatter
Metadata
Title
Differential Equations
Author
Christian Constanda
Copyright Year
2013
Publisher
Springer New York
Electronic ISBN
978-1-4614-7297-1
Print ISBN
978-1-4614-7296-4
DOI
https://doi.org/10.1007/978-1-4614-7297-1

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