Table of contents for A first course in differential equations with modeling applications / Dennis G. Zill.


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INTRODUCTION TO            DIFFERENTIAL EQUATIONS                                            I
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
3   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     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
5     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
6     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
CHAPTER 6 IN REVIEW     253
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
SSYSTEMS 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
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
CHAPTER 9 IN REVIEW     362
APPENDICES
I    Gamma Function    APP-1
II   Matrices   APP-3
III  Laplace Transforms  APP-21
Answers for Selected Odd-Numbered Problems  ANS-1



Library of Congress subject headings for this publication: Differential equations Textbooks