Elementary Differential Equations with Boundary Value Problems

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Acknowledgments......Page 14
Contents......Page 6
Preface......Page 12
1.2 Examples of Di.erential Equations......Page 18
1.3 Direction Fields......Page 25
2.1 Introduction......Page 32
2.2 First Order Linear Di.erential Equations......Page 36
2.3 Introduction to Mathematical Models......Page 48
2.4 Population Dynamics and Radioactive Decay......Page 58
2.5 First Order Nonlinear Di.erential Equations......Page 65
2.6 Separable First Order Equations......Page 71
2.7 Exact Differential Equations......Page 80
2.8 The Logistic Population Model......Page 87
2.9 Applications to Mechanics......Page 94
2.10 Euler’s Method......Page 106
Review Exercises......Page 117
Projects......Page 118
Chapter 3 Second and Higher Order Linear Differential Equations......Page 124
3.1 Introduction......Page 125
3.2 The General Solution of Homogeneous Equations......Page 132
3.3 Constant Coe.cient Homogeneous Equations......Page 138
3.4 Real Repeated Roots; Reduction of Order......Page 144
3.5 Complex Roots......Page 149
3.6 Unforced Mechanical Vibrations......Page 159
3.7 The General Solution of a Linear Nonhomogeneous Equation......Page 171
3.8 The Method of Undetermined Coe.cients......Page 175
3.9 The Method of Variation of Parameters......Page 185
3.10 Forced Mechanical Vibrations, Electrical Networks, and Resonance......Page 191
3.11 Higher Order Linear Homogeneous Di.erential Equations......Page 205
3.12 Higher Order Homogeneous Constant Coe.cient Di.erential Equations......Page 212
3.13 Higher Order Linear Nonhomogeneous Di.erential Equations......Page 218
Projects......Page 223
4.1 Introduction......Page 230
4.2 Existence and Uniqueness......Page 240
4.3 Homogeneous Linear Systems......Page 245
4.4 Constant Coe.cient Homogeneous Systems; the Eigenvalue Problem......Page 255
4.5 Real Eigenvalues and the Phase Plane......Page 264
4.6 Complex Eigenvalues......Page 273
4.7 Repeated Eigenvalues......Page 283
4.8 Nonhomogeneous Linear Systems......Page 294
4.9 Numerical Methods for Systems of Linear Di.erential Equations......Page 305
4.10 The Exponential Matrix and Diagonalization......Page 317
Review Exercises......Page 327
Projects......Page 328
5.1 Introduction......Page 334
5.2 Laplace Transform Pairs......Page 346
5.3 The Method of Partial Fractions......Page 361
5.4 Laplace Transforms of Periodic Functions and System Transfer Functions......Page 367
5.5 Solving Systems of Di.erential Equations......Page 376
5.6 Convolution......Page 385
5.7 The Delta Function and Impulse Response......Page 394
Projects......Page 402
6.1 Introduction......Page 408
6.2 Equilibrium Solutions and Direction Fields......Page 417
6.3 Conservative Systems......Page 430
6.4 Stability......Page 441
6.5 Linearization and the Local Picture......Page 450
6.6 Two-Dimensional Linear Systems......Page 465
6.7 Predator-Prey Population Models......Page 475
Projects......Page 483
7.1 Introduction......Page 488
7.2 Euler’s Method, Heun’s Method, and the Modi.ed Euler’s Method......Page 490
7.3 Taylor Series Methods......Page 496
7.4 Runge-Kutta Methods......Page 510
Appendix 1: Convergence of One-Step Methods......Page 523
Appendix 2: Stability of One-Step Methods......Page 524
Projects......Page 527
8.1 Introduction......Page 532
8.2 Series Solutions Near an Ordinary Point......Page 544
8.3 The Euler Equation......Page 553
8.4 Solutions Near a Regular Singular Point and the Method of Frobenius......Page 559
8.5 The Method of Frobenius Continued: Special Cases and a Summary......Page 567
Projects......Page 578
9.1 Introduction......Page 582
9.2 Heat Flow in a Thin Bar; Separation of Variables......Page 587
9.3 Series Solutions......Page 597
9.4 Calculating the Solution......Page 606
9.5 Fourier Series......Page 617
9.6 The Wave Equation......Page 633
9.7 Laplace’s Equation......Page 645
9.8 Higher-Dimensional Problems; Nonhomogeneous Equations......Page 658
Project......Page 672
10.1 Introduction......Page 676
10.2 The Cauchy Problem......Page 679
10.3 Existence and Uniqueness......Page 685
10.4 The Method of Characteristics......Page 688
Projects......Page 696
11.1 Introduction......Page 698
11.2 Existence and Uniqueness......Page 699
11.3 Two-Point Boundary Value Problems for Linear Systems......Page 710
11.4 Sturm-Liouville Boundary Value Problems......Page 722
Project......Page 732
Answers......Page 735
C......Page 788
E......Page 789
F......Page 790
I......Page 791
M......Page 792
P......Page 793
R......Page 794
T......Page 795
Z......Page 796