Greenberg Ordinary Differential Equations

1. Auflage 2014
ISBN: 978-1-118-24338-1
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 560 Seiten, E-Book

Features a balance between theory, proofs, and examples andprovides applications across diverse fields of study
Ordinary Differential Equations presents a thorough discussionof first-order differential equations and progresses to equationsof higher order. The book transitions smoothly from first-order tohigher-order equations, allowing readers to develop a completeunderstanding of the related theory.
Featuring diverse and interesting applications from engineering,bioengineering, ecology, and biology, the book anticipatespotential difficulties in understanding the various solution stepsand provides all the necessary details. Topical coverageincludes:
* First-Order Differential Equations
* Higher-Order Linear Equations
* Applications of Higher-Order Linear Equations
* Systems of Linear Differential Equations
* Laplace Transform
* Series Solutions
* Systems of Nonlinear Differential Equations
In addition to plentiful exercises and examples throughout, eachchapter concludes with a summary that outlines key concepts andtechniques. The book's design allows readers to interact with thecontent, while hints, cautions, and emphasis are uniquely featuredin the margins to further help and engage readers.
Written in an accessible style that includes all needed detailsand steps, Ordinary Differential Equations is an excellent book forcourses on the topic at the upper-undergraduate level. The bookalso serves as a valuable resource for professionals in the fieldsof engineering, physics, and mathematics who utilize differentialequations in their everyday work.
An Instructors Manual is available upon request. Email [e-mail] forinformation. There is also a Solutions Manual available. The ISBNis 9781118398999.

Autoren/Hrsg.

Greenberg, Michael D.

Inhaltsverzeichnis

Preface viii
1. First-Order Differential Equations 1
1.1 Motivation and Overview 1
1.2 Linear First-Order Equations 11
1.3 Applications of Linear First-Order Equations 24
1.4 Nonlinear First-Order Equations That Are Separable 43
1.5 Existence and Uniqueness 50
1.6 Applications of Nonlinear First-Order 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. Higher-Order Linear Equations 99
2.1 Linear Differential Equations of Second Order 99
2.2 Constant-Coefficient Equations 103
2.3 Complex Roots 113
2.4 Linear Independence; Existence, Uniqueness, General Solution118
2.5 Reduction of Order 128
2.6 Cauchy-Euler Equations 134
2.7 The General Theory for Higher-Order 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 Higher-Order 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 Steady-State Diffusion; A Boundary Value Problem 202
3.6 Introduction to the Eigenvalue Problem; Column Buckling211
4. Systems of Linear Differential Equations 219
4.1 Introduction, and Solution by Elimination 219
4.2 Application to Coupled Oscillators 230
4.3 N-Space 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 MatrixExponential 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 Equations334
5.4 Discontinuous Forcing Functions; Heaviside Step Function347
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 Method468
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 Plane497
Answers to Exercises 501

Über Autor(innen)

MICHAEL D. GREENBERG, PhD, is Professor Emeritus of Mechanical Engineering at the University of Delaware where he teaches courses on engineering mathematics and is a three-time recipient of the University of Delaware Excellence in Teaching Award. Greenberg's research has emphasized vortex methods in aerodynamics and hydrodynamics.