1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1
1.1 Definitions and Terminology 2
1.2 Initial-Value Problems 15
1.3 Differential Equations as Mathematical Models 22
Chapter 1 in Review 37
2 FIRST-ORDER DIFFERENTIAL EQUATIONS 39
2.1 Solution Curves Without the Solution 40
2.2 Separable Variables 51
2.3 Linear Equations 60
2.4 Exact Equations 72
2.5 Solutions by Substitutions 80
2.6 A Numerical Solution 86
Chapter 2 in Review 92
3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 95
3.1 Linear Equations 96
3.2 Nonlinear Equations 109
3.3 Systems of Linear and Nonlinear Differential Equations 121
Chapter 3 in Review 130
Project Module:Harvesting of Renewable Natural Resources,by Gilbert N.Lewis 133
4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 138
4.1 Preliminary Theory:Linear Equations 139
4.1.1 Initial-Value and Boundary-Value Problems 139
4.1.2 Homogeneous Equations 142
4.1.3 Nonhomogeneous Equations 148
4.2 Reduction of Order 154
4.3 Homogeneous Linear Equations with Constant Coefficients 158
4.4 Undetermined Coefficients—Superposition Approach 167
4.5 Undetermined Coefficients—Annihilator Approach 178
4.6 Variation of Parameters 188
4.7 Cauchy-Euler Equation 193
4.8 Solving Systems of Linear Equations by Elimination 201
4.9 Nonlinear Equations 207
Chapter 4 in Review 212
5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 215
5.1 Linear Equations:Initial-Value Problems 216
5.1.1 Spring/Mass Systems:Free Undamped Motion 216
5.1.2 Spring/Mass Systems:Free Damped Motion 220
5.1.3 Spring/Mass Systems:Driven Motion 224
5.1.4 Series Circuit Analogue 227
5.2 Linear Equations:Boundary-Value Problems 237
5.3 Nonlinear Equations 247
Chapter 5 in Review 259
Project Module:The Collapse of the Tacoma Narrows Suspension Bridge,by Gilbert N.Lewis 263
6 SERIES SOLUTIONS OF LINEAR EQUATIONS 267
6.1 Solutions About Ordinary Points 268
6.1.1 Review of Power Series 268
6.1.2 Power Series Solutions 271
6.2 Solutions About Singular Points 280
6.3 Two Special Equations 292
Chapter 6 in Review 304
7 THE LAPLACE TRANSFORM 306
7.1 Definition of the Laplace Transform 307
7.2 Inverse Transform and Transforms of Derivatives 314
7.3 Translation Theorems 324
7.3.1 Translation on the s-Axis 324
7.3.2 Translation on the t-Axis 328
7.4 Additional Operational Properties 338
7.5 Dirac Delta Function 351
7.6 Systems of Linear Equations 354
Chapter 7 in Review 361
8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 364
8.1 Preliminary Theory 365
8.2 Homogeneous Linear Systems with Constant Coefficients 375
8.2.1 Distinct Real Eigenvalues 376
8.2.2 Repeated Eigenvalues 380
8.2.3 Complex Eigenvalues 384
8.3 Variation of Parameters 393
8.4 Matrix Exponential 399
Chapter 8 in Review 404
Project Module:Earthquake Shaking of Multistory Buildings,by Gilbert N.Lewis 406
9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 410
9.1 Euler Methods and Error Analysis 411
9.2 Runge-Kutta Methods 417
9.3 Multistep Methods 424
9.4 Higher-Order Equations and Systems 427
9.5 Second-Order Boundary-Value Problems 433
Chapter 9 in Review 438