Differential Equations and Their Applications: An Introduction to Applied Mathematics

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Springer Science & Business Media, Dec 5, 1992 - Mathematics - 578 pages
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Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as weil as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high Ievel of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. T AM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Seiences ( AMS) series, which will focus on advanced textbooks and research Ievel monographs. Preface to the Fourth Edition There are two major changes in the Fourth Edition of Differential Equations and Their Applications. The first concerns the computer programs in this text. In keeping with recent trends in computer science, we have replaced all the APL programs with Pascal and C programs. The Pascal programs appear in the text in place ofthe APL programs, where they are followed by the Fortran programs, while the C programs appear in Appendix C.
 

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This book is not very organized. It is hard to find different sections of chapters because there are no breaks in between. It explains things VERY well if you already know what is going on, but before I understood differential equations a little bit (I used Schaum's) it was rather difficult. It has some fascinating examples of how differential equations are applied and is over all a good book, it is just somewhat difficult to learn from. 

Contents

Firstorder differential equations
1
12 Firstorder linear differential equations
2
13 The Van Meegeren art forgeries
11
14 Separable equations
20
15 Population models
26
16 The spread of technological innovations
39
17 An atomic waste disposal problem
46
18 The dynamics of tumor growth mixing problems and orthogonal trajectories
52
33 Dimension of a vector space
279
34 Applications of linear algebra to differential equations
291
35 The theory of determinants
297
36 Solutions of simultaneous linear equations
310
37 Linear transformations
320
38 The eigenvalueeigenvector method of finding solutions
333
39 Complex roots
341
310 Equal roots
345

19 Exact equations and why we cannot solve very many differential equations
58
110 The existenceuniqueness theorem Picard iteration
67
111 Finding roots of equations by iteration
81
1111 Newtons method
87
112 Difference equations and how to compute the interest due on your student loans
91
113 Numerical approximations Eulers method
96
1131 Error analysis for Eulers method
100
114 The three term Taylor series method
107
115 An improved Euler method
109
116 The RungeKutta method
112
117 What to do in practice
116
Secondorder linear differential equations
127
22 Linear equations with constant coefficients
138
221 Complex roots
141
222 Equal roots reduction of order
145
23 The nonhomogeneous equation
151
24 The method of variation of parameters
153
25 The method of judicious guessing
157
26 Mechanical vibrations
165
261 The Tacoma Bridge disaster
173
262 Electrical networks
175
27 A model for the detection of diabetes
178
28 Series solutions
185
257 Singular points Euler equations
198
282 Regular singular points the method of Frobenius
203
283 Equal roots and roots differing by an integer
219
29 The method of Laplace transforms
225
210 Some useful properties of Laplace transforms
233
211 Differential equations with discontinuous righthand sides
238
212 The Dirac delta function
243
213 The convolution integral Consider the initialvalue problem
251
214 The method of elimination for systems
257
215 Higherorder equations
259
Systems of differential equations
264
32 Vector spaces
273
311 Fundamental matrix solutions eAl
355
312 The nonhomogeneous equation variation of parameters
360
313 Solving systems by Laplace transforms
368
Qualitative theory of differential equations
372
42 Stability of linear systems
378
43 Stability of equilibrium solutions
385
44 The phaseplane
394
45 Mathematical theories of war
398
452 Lanchesters combat models and the battle of Iwo Jima
405
46 Qualitative properties of orbits
414
47 Phase portraits of linear systems
418
48 Long time behavior of solutions the PoincareBendixson Theorem
428
49 Introduction to bifurcation theory
437
410 Predatorprey problems or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I
443
411 The principle of competitive exclusion in population biology
451
412 The Threshold Theorem of epidemiology
458
413 A model for the spread of gonorrhea
465
Separation of variables and Fourier series
476
52 Introduction to partial differential equations
481
53 The heat equation separation of variables
483
54 Fourier series
487
55 Even and odd functions
493
56 Return to the heat equation
498
57 The wave equation
503
58 Laplaces equation
508
SturmLiouville boundary value problems
514
62 Inner product spaces
515
63 Orthogonal bases Hermitian operators
526
64 SturmLiouville theory
533
Some simple facts concerning functions of several variables
545
Sequences and series
547
C Programs
549
Answers to oddnumbered exercises
557
Index
575
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