Differential Equations: A Primer for Scientists and Engineers (Springer Undergraduate Texts in Mathematics and Technology)

✍ Scribed by Christian Constanda

• Publisher: Springer
• Year: 2017
• Tongue: English
• Leaves: 300
• Series: Springer Undergraduate Texts in Mathematics and Technology
• Edition: 2nd ed. 2017
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface to the Second Edition
Preface to the First Edition
Contents
Acronyms
1 Introduction
1.1 Calculus Prerequisites
1.2 Differential Equations and Their Solutions
1.3 Initial and Boundary Conditions
1.4 Classification of Differential Equations
2 First-Order Equations
2.1 Separable Equations
2.2 Linear Equations
2.3 Homogeneous Polar Equations
2.4 Bernoulli Equations
2.5 Riccati Equations
2.6 Exact Equations
2.7 Existence and Uniqueness Theorems
2.8 Direction Fields
3 Mathematical Models with First-Order Equations
3.1 Models with Separable Equations
3.2 Models with Linear Equations
3.3 Autonomous Equations
4 Linear Second-Order Equations
4.1 Mathematical Models with Second-Order Equations
4.2 Algebra Prerequisites
4.3 Homogeneous Equations
4.3.1 Initial Value Problems
4.3.2 Boundary Value Problems
4.4 Homogeneous Equations with Constant Coefficients
4.4.1 Real and Distinct Characteristic Roots
4.4.2 Repeated Characteristic Roots
4.4.3 Complex Conjugate Characteristic Roots
4.5 Nonhomogeneous Equations
4.5.1 Method of Undetermined Coefficients: Simple Cases
4.5.2 Method of Undetermined Coefficients: General Case
4.5.3 Method of Variation of Parameters
4.6 Cauchy–Euler Equations
4.7 Nonlinear Equations
5 Mathematical Models with Second-Order Equations
5.1 Free Mechanical Oscillations
5.1.1 Undamped Free Oscillations
5.1.2 Damped Free Oscillations
5.2 Forced Mechanical Oscillations
5.2.1 Undamped Forced Oscillations
5.2.2 Damped Forced Oscillations
5.3 Electrical Vibrations
6 Higher-Order Linear Equations
6.1 Modeling with Higher-Order Equations
6.2 Algebra Prerequisites
6.2.1 Matrices and Determinants of Higher Order
6.2.2 Systems of Linear Algebraic Equations
6.2.3 Linear Independence and the Wronskian
6.3 Homogeneous Differential Equations
6.4 Nonhomogeneous Equations
6.4.1 Method of Undetermined Coefficients
6.4.2 Method of Variation of Parameters
7 Systems of Differential Equations
7.1 Modeling with Systems of Equations
7.2 Algebra Prerequisites
7.2.1 Operations with Matrices
7.2.2 Linear Independence and the Wronskian
7.2.3 Eigenvalues and Eigenvectors
7.3 Systems of First-Order Differential Equations
7.4 Homogeneous Linear Systems with Constant Coefficients
7.4.1 Real and Distinct Eigenvalues
7.4.2 Complex Conjugate Eigenvalues
7.4.3 Repeated Eigenvalues
7.5 Other Features of Homogeneous Linear Systems
7.6 Nonhomogeneous Linear Systems
8 The Laplace Transformation
8.1 Definition and Basic Properties
8.2 Further Properties
8.3 Solution of IVPs for Single Equations
8.3.1 Continuous Forcing Terms
8.3.2 Piecewise Continuous Forcing Terms
8.3.3 Forcing Terms with the Dirac Delta
8.3.4 Equations with Variable Coefficients
8.4 Solution of IVPs for Systems
9 Series Solutions
9.1 Power Series
9.2 Series Solution Near an Ordinary Point
9.3 Singular Points
9.4 Solution Near a Regular Singular Point
9.4.1 Distinct Roots That Do Not Differ by an Integer
9.4.2 Equal Roots
9.4.3 Distinct Roots Differing by an Integer
10 Numerical Methods
10.1 The Euler Method
10.2 The Euler Midpoint Method
10.3 The Improved Euler Method
10.4 The Runge–Kutta Method
Appendix
A Algebra Techniques
A.1 Partial Fractions
A.2 Synthetic Division
Appendix
B Calculus Techniques
B.1 Sign of a Function
B.2 Integration by Parts
B.3 Integration by Substitution
B.4 Overview of the Hyperbolic Functions
Appendix
C Table of Laplace Transforms
Appendix D The Greek Alphabet
Further Reading
Answers to Odd-Numbered Exercises
Index

✦ Subjects

Mathematics;Calculus; Differential equations