Preface to the Second Edition |
|
xiii | |
Preface to the First Edition |
|
xv | |
|
1 What Is a Differential Equation? |
|
|
1 | (62) |
|
|
1 | (4) |
|
1.2 A Taste of Ordinary Differential Equations |
|
|
5 | (1) |
|
1.3 The Nature of Solutions |
|
|
6 | (7) |
|
|
13 | (3) |
|
1.5 First-Order Linear Equations |
|
|
16 | (4) |
|
|
20 | (5) |
|
1.7 Orthogonal Trajectories and Families of Curves |
|
|
25 | (6) |
|
1.8 Homogeneous Equations |
|
|
31 | (3) |
|
|
34 | (4) |
|
|
38 | (5) |
|
1.10.1 Dependent Variable Missing |
|
|
39 | (1) |
|
1.10.2 Independent Variable Missing |
|
|
40 | (3) |
|
1.11 The Hanging Chain and Pursuit Curves |
|
|
43 | (9) |
|
|
43 | (5) |
|
|
48 | (4) |
|
|
52 | (11) |
|
Anatomy of an Application |
|
|
56 | (4) |
|
Problems for Review and Discovery |
|
|
60 | (3) |
|
2 Second-Order Linear Equations |
|
|
63 | (56) |
|
2.1 Second-Order Linear Equations with Constant Coefficients |
|
|
63 | (6) |
|
2.2 The Method of Undetermined Coefficients |
|
|
69 | (4) |
|
2.3 The Method of Variation of Parameters |
|
|
73 | (5) |
|
2.4 The Use of a Known Solution to Find Another |
|
|
78 | (3) |
|
2.5 Vibrations and Oscillations |
|
|
81 | (11) |
|
2.5.1 Undamped Simple Harmonic Motion |
|
|
82 | (2) |
|
|
84 | (3) |
|
|
87 | (2) |
|
2.5.4 A Few Remarks about Electricity |
|
|
89 | (3) |
|
2.6 Newton's Law of Gravitation and Kepler's Laws |
|
|
92 | (12) |
|
2.6.1 Kepler's Second Law |
|
|
95 | (2) |
|
|
97 | (3) |
|
|
100 | (4) |
|
2.7 Higher-Order Equations |
|
|
104 | (15) |
|
|
110 | (2) |
|
Anatomy of an Application |
|
|
112 | (4) |
|
Problems for Review and Discovery |
|
|
116 | (3) |
|
3 Power Series Solutions and Special Functions |
|
|
119 | (52) |
|
3.1 Introduction and Review of Power Series |
|
|
119 | (10) |
|
3.1.1 Review of Power Series |
|
|
120 | (9) |
|
3.2 Series Solutions of First-Order Equations |
|
|
129 | (5) |
|
3.3 Second-Order Linear Equations: Ordinary Points |
|
|
134 | (8) |
|
3.4 Regular Singular Points |
|
|
142 | (6) |
|
3.5 More on Regular Singular Points |
|
|
148 | (8) |
|
3.6 Gauss's Hypergeometric Equation |
|
|
156 | (15) |
|
|
161 | (2) |
|
|
163 | (2) |
|
Anatomy of an Application |
|
|
165 | (3) |
|
Problems for Review and Discovery |
|
|
168 | (3) |
|
|
171 | (24) |
|
|
172 | (1) |
|
|
173 | (4) |
|
|
177 | (3) |
|
4.4 An Improved Euler Method |
|
|
180 | (4) |
|
4.5 The Runge--Kutta Method |
|
|
184 | (11) |
|
Anatomy of an Application |
|
|
189 | (3) |
|
Problems for Review and Discovery |
|
|
192 | (3) |
|
5 Fourier Series: Basic Concepts |
|
|
195 | (44) |
|
|
195 | (9) |
|
5.2 Some Remarks about Convergence |
|
|
204 | (6) |
|
5.3 Even and Odd Functions: Cosine and Sine Series |
|
|
210 | (5) |
|
5.4 Fourier Series on Arbitrary Intervals |
|
|
215 | (4) |
|
|
219 | (20) |
|
|
224 | (2) |
|
Anatomy of an Application |
|
|
226 | (10) |
|
Problems for Review and Discovery |
|
|
236 | (3) |
|
6 Sturm--Liouville Problems and Boundary Value Problems |
|
|
239 | (30) |
|
6.1 What is a Sturm--Liouville Problem? |
|
|
239 | (7) |
|
6.2 Analyzing a Sturm--Liouville Problem |
|
|
246 | (4) |
|
6.3 Applications of the Sturm--Liouville Theory |
|
|
250 | (6) |
|
6.4 Singular Sturm--Liouville |
|
|
256 | (13) |
|
Anatomy of an Application |
|
|
263 | (4) |
|
Problems for Review and Discovery |
|
|
267 | (2) |
|
7 Partial Differential Equations and Boundary Value Problems |
|
|
269 | (36) |
|
7.1 Introduction and Historical Remarks |
|
|
269 | (4) |
|
7.2 Eigenvalues, Eigenfunctions, and the Vibrating String |
|
|
273 | (9) |
|
7.2.1 Boundary Value Problems |
|
|
273 | (1) |
|
7.2.2 Derivation of the Wave Equation |
|
|
274 | (3) |
|
7.2.3 Solution of the Wave Equation |
|
|
277 | (5) |
|
|
282 | (6) |
|
7.4 The Dirichlet Problem for a Disc |
|
|
288 | (17) |
|
7.4.1 The Poisson Integral |
|
|
291 | (5) |
|
|
296 | (1) |
|
Historical Note: Dirichlet |
|
|
297 | (1) |
|
Problems for Review and Discovery |
|
|
298 | (3) |
|
Anatomy of an Application |
|
|
301 | (4) |
|
|
305 | (40) |
|
|
305 | (3) |
|
8.1 Applications to Differential Equations |
|
|
308 | (5) |
|
8.2 Derivatives and Integrals of Laplace Transforms |
|
|
313 | (6) |
|
|
319 | (8) |
|
8.3.1 Abel's Mechanics Problem |
|
|
322 | (5) |
|
8.4 The Unit Step and Impulse Functions |
|
|
327 | (18) |
|
|
335 | (1) |
|
Anatomy of an Application |
|
|
336 | (3) |
|
Problems for Review and Discovery |
|
|
339 | (6) |
|
9 Systems of First-Order Equations |
|
|
345 | (34) |
|
|
345 | (3) |
|
|
348 | (8) |
|
9.3 Homogeneous Linear Systems with Constant Coefficients |
|
|
356 | (7) |
|
9.4 Nonlinear Systems: Volterra's Predator-Prey Equations |
|
|
363 | (16) |
|
Anatomy of an Application |
|
|
370 | (5) |
|
Problems for Review and Discovery |
|
|
375 | (4) |
|
|
379 | (64) |
|
10.1 Some Motivating Examples |
|
|
380 | (1) |
|
|
380 | (5) |
|
10.3 Types of Critical Points: Stability |
|
|
385 | (9) |
|
10.4 Critical Points and Stability for Linear Systems |
|
|
394 | (11) |
|
10.5 Stability by Liapunov's Direct Method |
|
|
405 | (6) |
|
10.6 Simple Critical Points of Nonlinear Systems |
|
|
411 | (7) |
|
10.7 Nonlinear Mechanics: Conservative Systems |
|
|
418 | (6) |
|
10.8 Periodic Solutions: The Poincare--Bendixson Theorem |
|
|
424 | (19) |
|
Historical Note: Poincare |
|
|
434 | (2) |
|
Anatomy of an Application |
|
|
436 | (3) |
|
Problems for Review and Discovery |
|
|
439 | (4) |
Appendix: Review of Linear Algebra |
|
443 | (12) |
Bibliography |
|
455 | (4) |
Index |
|
459 | |