Ordinary Differential EquationsISBN: 9781118230022
544 pages
February 2012

Description
Ordinary Differential Equations presents a thorough discussion of firstorder differential equations and progresses to equations of higher order. The book transitions smoothly from firstorder to higherorder equations, allowing readers to develop a complete understanding of the related theory.
Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps and provides all the necessary details. Topical coverage includes:

FirstOrder Differential Equations

HigherOrder Linear Equations

Applications of HigherOrder Linear Equations

Systems of Linear Differential Equations

Laplace Transform

Series Solutions

Systems of Nonlinear Differential Equations
In addition to plentiful exercises and examples throughout, each chapter concludes with a summary that outlines key concepts and techniques. The book's design allows readers to interact with the content, while hints, cautions, and emphasis are uniquely featured in the margins to further help and engage readers.
Written in an accessible style that includes all needed details and steps, Ordinary Differential Equations is an excellent book for courses on the topic at the upperundergraduate level. The book also serves as a valuable resource for professionals in the fields of engineering, physics, and mathematics who utilize differential equations in their everyday work.
An Instructors Manual is available upon request. Email sfriedman@wiley.com for information. There is also a Solutions Manual available. The ISBN is 9781118398999.
Table of Contents
1. FirstOrder Differential Equations 1
1.1 Motivation and Overview 1
1.2 Linear FirstOrder Equations 11
1.3 Applications of Linear FirstOrder Equations 24
1.4 Nonlinear FirstOrder Equations That Are Separable 43
1.5 Existence and Uniqueness 50
1.6 Applications of Nonlinear FirstOrder Equations 59
1.7 Exact Equations and Equations That Can Be Made Exact 71
1.8 Solution by Substitution 81
1.9 Numerical Solution by Euler’s Method 87
2. HigherOrder Linear Equations 99
2.1 Linear Differential Equations of Second Order 99
2.2 ConstantCoefficient Equations 103
2.3 Complex Roots 113
2.4 Linear Independence; Existence, Uniqueness, General Solution 118
2.5 Reduction of Order 128
2.6 CauchyEuler Equations 134
2.7 The General Theory for HigherOrder Equations 142
2.8 Nonhomogeneous Equations 149
2.9 Particular Solution by Undetermined Coefficients 155
2.10 Particular Solution by Variation of Parameters 163
3. Applications of HigherOrder Equations 173
3.1 Introduction 173
3.2 Linear Harmonic Oscillator; Free Oscillation 174
3.3 Free Oscillation with Damping 186
3.4 Forced Oscillation 193
3.5 SteadyState Diffusion; A Boundary Value Problem 202
3.6 Introduction to the Eigenvalue Problem; Column Buckling 211
4. Systems of Linear Differential Equations 219
4.1 Introduction, and Solution by Elimination 219
4.2 Application to Coupled Oscillators 230
4.3 NSpace and Matrices 238
4.4 Linear Dependence and Independence of Vectors 247
4.5 Existence, Uniqueness, and General Solution 253
4.6 Matrix Eigenvalue Problem 261
4.7 Homogeneous Systems with Constant Coefficients 270
4.8 Dot Product and Additional Matrix Algebra 283
4.9 Explicit Solution of x’ = Ax and the Matrix Exponential Function 297
4.10 Nonhomogeneous Systems 307
5. Laplace Transform 317
5.1 Introduction 317
5.2 The Transform and Its Inverse 319
5.3 Applications to the Solution of Differential Equations 334
5.4 Discontinuous Forcing Functions; Heaviside Step Function 347
5.5 Convolution 358
5.6 Impulsive Forcing Functions; Dirac Delta Function 366
6. Series Solutions 379
6.1 Introduction 379
6.2 Power Series and Taylor Series 380
6.3 Power Series Solution About a Regular Point 387
6.4 Legendre and Bessel Equations 395
6.5 The Method of Frobenius 408
7. Systems of Nonlinear Differential Equations 423
7.1 Introduction 423
7.2 The Phase Plane 424
7.3 Linear Systems 435
7.4 Nonlinear Systems 447
7.5 Limit Cycles 463
7.6 Numerical Solution of Systems by Euler’s Method 468
Appendix A. Review of Partial Fraction Expansions 479
Appendix B. Review of Determinants 483
Appendix C. Review of Gauss Elimination 491
Appendix D. Review of Complex Numbers and the Complex Plane 497
Answers to Exercises 501
Author Information
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Reviews
“It is clearly written, well illustrated and it could be useful for applied mathematicians, physicists, engineers and other related professionals and also for students who are interested in the applications of ordinary differential equations.” (Zentralblatt MATH, 1 June 2013)
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