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- Differential equaion with boungdary value problems 6판
Dennis.G.Zill
Michael R.Cullen 저
pdf파일입니다.
- 본문일부/목차
- 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1
Preface xi
1.1 Definitions and Terminology 2
1.2 Initial-Value Problems 13
1.3 Differential Equations as Mathematical Models 19
CHAPTER 1 IN REVIEW 32
2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34
2.1 Solution Curves Without a Solution 35
2.1.1 Direction Fields 35
2.1.2 Autonomous First-Order DEs 37
2.2 Separable Variables 44
2.3 Linear Equations 53
2.4 Exact Equations 62
2.5 Solutions by Substitutions 70
2.6 A Numerical Method 75
CHAPTER 2 IN REVIEW 80
MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82
3.1 Linear Models 83
3.2 Nonlinear Models 94
3.3 Modeling with Systems of First-Order DEs 105
CHAPTER 3 IN REVIEW 113
4
vi ● CONTENTS
HIGHER-ORDER DIFFERENTIAL EQUATIONS 117
4.1 Preliminary Theory—Linear Equations 118
4.1.1 Initial-Value and Boundary-Value Problems 118
4.1.2 Homogeneous Equations 120
4.1.3 Nonhomogeneous Equations 125
4.2 Reduction of Order 130
4.3 Homogeneous Linear Equations with Constant Coefficients 133
4.4 Undetermined Coefficients—Superposition Approach 140
4.5 Undetermined Coefficients—Annihilator Approach 150
4.6 Variation of Parameters 157
4.7 Cauchy-Euler Equation 162
4.8 Solving Systems of Linear DEs by Elimination 169
4.9 Nonlinear Differential Equations 174
CHAPTER 4 IN REVIEW 178
MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181
5.1 Linear Models: Initial-Value Problems 182
5.1.1 Spring/Mass Systems: Free Undamped Motion 182
5.1.2 Spring/Mass Systems: Free Damped Motion 186
5.1.3 Spring/Mass Systems: Driven Motion 189
5.1.4 Series Circuit Analogue 192
5.2 Linear Models: Boundary-Value Problems 199
5.3 Nonlinear Models 207
CHAPTER 5 IN REVIEW 216
SERIES SOLUTIONS OF LINEAR EQUATIONS 219
6.1 Solutions About Ordinary Points 220
6.1.1 Review of Power Series 220
6.1.2 Power Series Solutions 223
6.2 Solutions About Singular Points 231
6.3 Special Functions 241
6.3.1 Bessel’s Equation 241
6.3.2 Legendre’s Equation 248
7 THE LAPLACE TRANSFORM 255
7.1 Definition of the Laplace Transform 256
7.2 Inverse Transforms and Transforms of Derivatives 262
7.2.1 Inverse Transforms 262
7.2.2 Transforms of Derivatives 265
7.3 Operational Properties I 270
7.3.1 Translation on the s-Axis 271
7.3.2 Translation on the t-Axis 274
7.4 Operational Properties II 282
7.4.1 Derivatives of a Transform 282
7.4.2 Transforms of Integrals 283
7.4.3 Transform of a Periodic Function 287
7.5 The Dirac Delta Function 292
7.6 Systems of Linear Differential Equations 295
CHAPTER 7 IN REVIEW 300
8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303
8.1 Preliminary Theory—Linear Systems 304
8.2 Homogeneous Linear Systems 311
8.2.1 Distinct Real Eigenvalues 312
8.2.2 Repeated Eigenvalues 315
8.2.3 Complex Eigenvalues 320
8.3 Nonhomogeneous Linear Systems 326
8.3.1 Undetermined Coefficients 326
8.3.2 Variation of Parameters 329
8.4 Matrix Exponential 334
CHAPTER 8 IN REVIEW 337
9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339
9.1 Euler Methods and Error Analysis 340
9.2 Runge-Kutta Methods 345
9.3 Multistep Methods 350
9.4 Higher-Order Equations and Systems 353
9.5 Second-Order Boundary-Value Problems 358
10 PLANE AUTONOMOUS SYSTEMS 363
10.1 Autonomous Systems 364
10.2 Stability of Linear Systems 370
10.3 Linearization and Local Stability 378
10.4 Autonomous Systems as Mathematical Models 388
CHAPTER 10 IN REVIEW 395
11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397
11.1 Orthogonal Functions 398
11.2 Fourier Series 403
11.3 Fourier Cosine and Sine Series 408
11.4 Sturm-Liouville Problem 416
11.5 Bessel and Legendre Series 423
11.5.1 Fourier-Bessel Series 424
11.5.2 Fourier-Legendre Series 427
CHAPTER 11 IN REVIEW 430
12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432
12.1 Separable Partial Differential Equations 433
12.2 Classical PDEs and Boundary-Value Problems 437
12.3 Heat Equation 443
12.4 Wave Equation 445
12.5 Laplace’s Equation 450
12.6 Nonhomogeneous Boundary-Value Problems 455
12.7 Orthogonal Series Expansions 461
12.8 Higher-Dimensional Problems 466
CHAPTER 12
13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 471
13.1 Polar Coordinates 472
13.2 Polar and Cylindrical Coordinates 477
13.3 Spherical Coordinates 483
CHAPTER 13 IN REVIEW 486
14 INTEGRAL TRANSFORMS 488
14.1 Error Function 489
14.2 Laplace Transform 490
14.3 Fourier Integral 498
14.4 Fourier Transforms 504
CHAPTER 14 IN REVIEW 510
15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 511
15.1 Laplace’s Equation 512
15.2 Heat Equation 517
15.3 Wave Equation 522
CHAPTER 15 IN REVIEW 526
APPENDICES
I Gamma Function APP-1
II Matrices APP-3
III Laplace Transforms APP-21
Answers for Selected Odd-Numbered Problems ANS-1
Index I
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