Ordinary Differential Equations (eBook)

Michael D. Greenberg (Autor)

eBook Download: PDF

2014
John Wiley & Sons (Verlag)
978-1-118-24338-1 (ISBN)

Infos

Features a balance between theory, proofs, and examples and provides applications across diverse fields of study

Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order 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:

• 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, 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 upper-undergraduate 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 [e-mail] for information. There is also a Solutions Manual available. The ISBN is 9781118398999.

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.

Inhalt

Cover 1
Title Page 5
Copyright 6
Contents 7
Preface 15
1 First-Order Differential Equations 21
1.1 Motivation And Overview 21
1.1.1 Introduction 21
1.1.2 Modeling 23
1.1.3 The order of a differential equation 24
1.1.4 Linear and nonlinear equations 25
1.1.5 Ourplan 26
1.1.6 Direction field 27
1.1.7 Computer software 28
1.2 Linear First-Order Equations 31
1.2.1 The simplest case 31
1.2.2 The homogeneous equation 32
1.2.3 Solving the full equation by the integrating factor method 34
1.2.4 Existence and uniqueness for the linear equation 37
1.3 Applications Of Linear First-Order Equations 44
1.3.1 Population dynamics exponential model
1.3.2 Radioactive decay carbon dating
1.3.3 Mixing problems a one-compartment model
1.3.4 The phase line, equilibrium points, and stability 50
1.3.5 Electrical circuits 51
1.4 Nonlinear First-Order Equations That Are Separable 63
1.5 Existence And Uniqueness 70
1.5.1 An existence and uniqueness theorem 70
1.5.2 Illustrating the theorem 71
1.5.3 Application to free fall physical significance of nonuniqueness
1.6 Applications Of Nonlinear First-Order Equations 79
1.6.1 The logistic model of population dynamics 79
1.6.2 Stability of equilibrium points and linearized stability analysis 81
1.7 Exact Equations And Equations That Can Be Made Exact 91
1.7.1 Exact differential equations 91
1.7.2 Making an equation exact integrating factors
1.8 Solution By Substitution 101
1.8.1 Bernoulli's equation 101
1.8.2 Homogeneous equations 103
1.9 Numerical Solution By Euler's Method 107
1.9.1 Euler's method 107
1.9.2 Convergence of Euler's method 110
1.9.3 Higher-order methods 112
Chapter 1 Review 115
2 Higher-Order Linear Equations 119
2.1 Linear Differential Equations Of Second Order 119
2.1.1 Introduction 119
2.1.2 Operator notation and linear differential operators 120
2.1.3 Superposition principle 121
2.2 Constant-Coefficient Equations 123
2.2.1 Constant coefficients 123
2.2.2 Seeking a general solution 124
2.2.3 Initial value problem 130
2.3 Complex Roots 133
2.3.1 Complex exponential function 133
2.3.2 Complex characteristic roots 135
2.4 Linear Independence Existence, Uniqueness, General Solution
2.4.1 Linear dependence and linear independence 139
2.4.2 Existence, uniqueness, and general solution 141
2.4.3 Abel's formula and Wronskian test for linear independence 144
2.4.4 Building a solution method on these results 145
2.5 Reduction Of Order 148
2.5.1 Deriving the formula 148
2.5.2 The method rather than the formula 151
2.5.3 About the method of reduction of order 152
2.6 Cauchy-Euler Equations 154
2.6.1 General solution 155
2.6.2 Repeated roots and reduction of order 156
2.6.3 Complex roots 158
2.7 The General Theory For Higher-Order Equations 162
2.7.1 Theorems for nth-order linear equations 163
2.7.2 Constant-coefficient equations 164
2.7.3 Cauchy-Euler equations 166
2.8 Nonhomogeneous Equations 169
2.8.1 General solution 169
2.8.2 The scaling and superposition of forcing functions 171
2.9 Particular Solution By Undetermined Coefficients 175
2.9.1 Undetermined coefficients 175
2.9.2 A special case the complex exponential method
2.10 Particular Solution By Variation Of Parameters 183
2.10.1 First-order equations 183
2.10.2 Second-order equations 185
Chapter 2 Review 190
3 Applications Of Higher-Order Equations 193
3.1 Introduction 193
3.2 Linear Harmonic Oscillator Free Oscillation
3.2.1 Mass-spring oscillator 194
3.2.2 Undamped free oscillation 196
3.2.3 Pendulum 199
3.3 Free Oscillation With Damping 206
3.3.1 Underdamped 207
3.3.2 Critically damped 208
3.3.3 Overdamped 208
3.4 Forced Oscillation 213
3.4.1 Undamped, c = 0 213
3.4.2 Damped, c > 0
3.5 Steady-State Diffusion A Boundary Value Problem
3.5.1 Boundary value problems existence and uniqueness
3.5.2 Steady-state heat conduction in a rod 223
3.6 Introduction To The Eigenvalue Problem Column Buckling
3.6.1 An eigenvalue problem 231
3.6.2 Application to column buckling 233
Chapter 3 Review 238
4 Systems Of Linear Differential Equations 239
4.1 Introduction, And Solution By Elimination 239
4.1.1 Introduction 239
4.1.2 Physical examples 240
4.1.3 Solutions, existence, and uniqueness 241
4.1.4 Solution by elimination 242
4.1.5 Auxiliary variables 245
4.2 Application To Coupled Oscillators 250
4.2.1 Coupled oscillators 250
4.2.2 Reduction to first-order system by auxiliary variables 251
4.2.3 The free vibration 251
4.2.4 The forced vibration 254
4.3 N-Space And Matrices 258
4.3.1 Passage from 2-space to n-space 258
4.3.2 Matrix operators on vectors in n-space 260
4.3.3 Identity matrix and zero matrix 262
4.3.4 Relevance to systems of linear algebraic equations 262
4.3.5 Vector and matrix functions 264
4.4 Linear Dependence And Independence Of Vectors 267
4.4.1 Linear dependence of a set of constant vectors in n-space 267
4.4.2 Linear dependence of vector functions in n-space 270
4.5 Existence, Uniqueness, And General Solution 273
4.5.1 The key theorems 273
4.5.2 Illustrating the theorems 277
4.6 Matrix Eigenvalue Problem 281
4.6.1 The eigenvalue problem 281
4.6.2 Solving an eigenvalue problem 282
4.6.3 Complex eigenvalues and eigenvectors 286
4.7 Homogeneous Systems With Constant Coefficients 290
4.7.1 Solution by the method of assumed exponential form 290
4.7.2 Application to the two-mass oscillator 294
4.7.3 The case of repeated eigenvalues 296
4.7.4 Modifying the method if there are defective eigenvalues 298
4.7.5 Complex eigenvalues 299
4.8 Dot Product And Additional Matrix Algebra 303
4.8.1 More about n-space: dot product, norm, and angle 303
4.8.2 Algebra of matrix operators 305
4.8.3 Inverse matrix 309
4.9 Explicit Solution Of x' = Ax And The Matrix Exponential Function 317
4.9.1 Matrix exponential solution 317
4.9.2 Getting the exponential matrix series into closed form 320
4.10 Nonhomogeneous Systems 327
4.10.1 Solution by variation of parameters 327
4.10.2 Constant coefficient matrix 330
4.10.3 Particular solution by undetermined coefficients 331
Chapter 4 Review 334
5 Laplace Transform 337
5.1 Introduction 337
5.2 The Transform And Its Inverse 338
5.2.1 Laplace transform 338
5.2.2 Linearity property of the transform 341
5.2.3 Exponential order, piecewise continuity, and conditions for existence of the transform 343
5.2.4 Inverse transform 346
5.2.5 Introduction to the determination of inverse transforms 347
5.3 Application To The Solution Of Differential Equations 354
5.3.1 First-order equations 354
5.3.2 Higher-order equations 356
5.3.3 Systems 359
5.3.4 Application to a nonconstant-coefficient equation Bessel's equation
5.4 Discontinuous Forcing Functions Heaviside Step Function
5.4.1 Motivation 367
5.4.2 Heaviside step function and piecewise-defined functions 367
5.4.3 Transforms of Heaviside and time-delayed functions 369
5.4.4 Differential equations with piecewise-defined forcing functions 371
5.4.5 Periodic forcing functions 373
5.5 Convolution 378
5.5.1 Definition of Laplace convolution 378
5.5.2 Convolution theorem 379
5.5.3 Applications 380
5.5.4 Integro-differential equations and integral equations 382
5.6 Impulsive Forcing Functions Dirac Delta Function
5.6.1 Impulsive forces 386
5.6.2 Dirac delta function 388
5.6.3 The jump caused by the delta function 390
5.6.4 Caution 391
5.6.5 Impulse response function 392
Chapter 5 Review 396
6 Series Solutions 399
6.1 Introduction 399
6.2 Power Series And Taylor Series 400
6.2.1 Power series 400
6.2.2 Manipulation of power series 402
6.2.3 Taylor series 403
6.3 Power Series Solution About A Regular Point 407
6.3.1 Power series solution theorem 407
6.3.2 Applications 409
6.4 Legendre And Bessel Equations 415
6.4.1 Introduction 415
6.4.2 Legendre's equation 415
6.4.3 Bessel's equation 419
6.5 The Method of Frobenius 428
6.5.1 Motivation 428
6.5.2 Regular and irregular singular points 429
6.5.3 The method of Frobenius 430
Chapter 6 Review 440
7 Systems Of Nonlinear Differential Equations 443
7.1 Introduction 443
7.2 The Phase Plane 444
7.2.1 Phase plane method 444
7.2.2 Application to nonlinear pendulum 447
7.2.3 Singular points and their stability 451
7.3 Linear Systems 455
7.3.1 Introduction 455
7.3.2 Purely imaginary eigenvalues (Center) 457
7.3.3 Complex conjugate eigenvalues (Spiral) 458
7.3.4 Real eigenvalues of the same sign (Node) 459
7.3.5 Real eigenvalues of opposite sign (Saddle) 463
7.4 Nonlinear Systems 467
7.4.1 Local linearization 468
7.4.2 Predator-prey population dynamics 470
7.4.3 Competing species 473
7.5 Limit Cycles 483
7.6 Numerical Solution Of Systems By Euler's Method 488
7.6.1 Initial value problems 488
7.6.2 Existence and uniqueness for nonlinear systems 491
7.6.3 Linear boundary value problems 492
Chapter 7 Review 496
Appendix A: Review Of Partial Fraction Expansions 499
Appendix B: Review Of Determinants 503
Appendix C: Review Of Gauss Elimination 511
Appendix D: Review Of Complex Numbers And The Complex Plane 517
Answers To Exercises 521
Index 541
Selected formulas 546

Rezensionen

"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)

Produkt-Details

• Erscheint lt. Verlag: 30.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Analysis • Applied Mathematics in Engineering • Applied Mathmatics in Engineering • Differential Equations • Differentialgleichung • Differentialgleichungen • Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • Mathematics • Mathematik • Mathematik in den Ingenieurwissenschaften • Numerical Methods & Algorithms • Numerische Methoden u. Algorithmen • Ordinary differential equations, first order euqations, second order equations, constant coefficients, fourth order equations, linear equations, nonlinear equations, Maple, Mathematica, MATLAB, Laplace transform, higher-order equations, partial fraction expansions, Gauss elimination
• ISBN-10: 1-118-24338-2 / 1118243382
• ISBN-13: 978-1-118-24338-1 / 9781118243381