A First Course in Differential Equations with Modeling Applications 10e

A First Course in Differential Equations with Modeling Applications 10e
اسم المؤلف
DENNIS G. ZILL
التاريخ
29 يناير 2016
التصنيف
المشاهدات
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A First Course in Differential Equations with Modeling Applications 10e
with Modeling Applications
DENNIS G. ZILL
Loyola Marymount University
Contents
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1
Preface ix
Projects P-1
1.1 Definitions and Terminology 2
1.2 Initial-Value Problems 13
1.3 Differential Equations as Mathematical Models 20
Chapter 1 in Review 33
2 FIRST-ORDER DIFFERENTIAL EQUATIONS 35
2.1 Solution Curves Without a Solution 36
2.1.1 Direction Fields 36
2.1.2 Autonomous First-Order DEs 38
2.2 Separable Equations 46
2.3 Linear Equations 54
2.4 Exact Equations 63
2.5 Solutions by Substitutions 71
2.6 A Numerical Method 75
Chapter 2 in Review 80
MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 83
3.1 Linear Models 84
3.2 Nonlinear Models 95
3.3 Modeling with Systems of First-Order DEs 106
Chapter 3 in Review 113
HIGHER-ORDER DIFFERENTIAL EQUATIONS 116
4.1 Preliminary Theory—Linear Equations 117
4.1.1 Initial-Value and Boundary-Value Problems 117
4.1.2 Homogeneous Equations 119
4.1.3 Nonhomogeneous Equations 124
4.2 Reduction of Order 129
4.3 Homogeneous Linear Equations with Constant Coefficients 132
4.4 Undetermined Coefficients—Superposition Approach 139
4.5 Undetermined Coefficients—Annihilator Approach 149
4.6 Variation of Parameters 156
4.7 Cauchy-Euler Equation 162
4.8 Green’s Functions 169
4.8.1 Initial-Value Problems 169
4.8.2 Boundary-Value Problems 176
4.9 Solving Systems of Linear DEs by Elimination 180
4.10 Nonlinear Differential Equations 185
Chapter 4 in Review 190
MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 192
5.1 Linear Models: Initial-Value Problems 193
5.1.1 Spring/Mass Systems: Free Undamped Motion 193
5.1.2 Spring/Mass Systems: Free Damped Motion 197
5.1.3 Spring/Mass Systems: Driven Motion 200
5.1.4 Series Circuit Analogue 203
5.2 Linear Models: Boundary-Value Problems 210
5.3 Nonlinear Models 218
Chapter 5 in Review 228
SERIES SOLUTIONS OF LINEAR EQUATIONS 231
6.1 Review of Power Series 232
6.2 Solutions About Ordinary Points 238
6.3 Solutions About Singular Points 247
6.4 Special Functions 257
Chapter 6 in Review 271
7 THE LAPLACE TRANSFORM 273
7.1 Definition of the Laplace Transform 274
7.2 Inverse Transforms and Transforms of Derivatives 281
7.2.1 Inverse Transforms 281
7.2.2 Transforms of Derivatives 284
7.3 Operational Properties I 289
7.3.1 Translation on the s-Axis 290
7.3.2 Translation on the t-Axis 293
7.4 Operational Properties II 301
7.4.1 Derivatives of a Transform 301
7.4.2 Transforms of Integrals 302
7.4.3 Transform of a Periodic Function 307
7.5 The Dirac Delta Function 312
7.6 Systems of Linear Differential Equations 315
Chapter 7 in Review 320
8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 325
8.1 Preliminary Theory—Linear Systems 326
8.2 Homogeneous Linear Systems 333
8.2.1 Distinct Real Eigenvalues 334
8.2.2 Repeated Eigenvalues 337
8.2.3 Complex Eigenvalues 342
8.3 Nonhomogeneous Linear Systems 348
8.3.1 Undetermined Coefficients 348
8.3.2 Variation of Parameters 351
8.4 Matrix Exponential 356
Chapter 8 in Review 360
9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 362
9.1 Euler Methods and Error Analysis 363
9.2 Runge-Kutta Methods 368
9.3 Multistep Methods 373
9.4 Higher-Order Equations and Systems 375
9.5 Second-Order Boundary-Value Problems 380
Chapter 9 in Review 384
I Gamma Function APP-1
II Matrices APP-3
III Laplace Transforms APP-21
Answers for Selected Odd-Numbered Problems ANS-1
Index I-1
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