Differential Equations: An Introduction with Mathematica®

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Springer Science & Business Media, Aug 3, 2004 - Mathematics - 434 pages
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Goals and Emphasis of the Book Mathematicians have begun to find productive ways to incorporate computing power into the mathematics curriculum. There is no attempt here to use computing to avoid doing differential equations and linear algebra. The goal is to make some first ex plorations in the subject accessible to students who have had one year of calculus. Some of the sciences are now using the symbol-manipulative power of Mathemat ica to make more of their subject accessible. This book is one way of doing so for differential equations and linear algebra. I believe that if a student's first exposure to a subject is pleasant and exciting, then that student will seek out ways to continue the study of the subject. The theory of differential equations and of linear algebra permeates the discussion. Every topic is supported by a statement of the theory. But the primary thrust here is obtaining solutions and information about solutions, rather than proving theorems. There are other courses where proving theorems is central. The goals of this text are to establish a solid understanding of the notion of solution, and an appreciation for the confidence that the theory gives during a search for solutions. Later the student can have the same confidence while personally developing the theory.
 

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Contents

About Differential Equations
1
11 Numerical Methods
8
12 Uniqueness Considerations
17
13 Differential Inclusions Optional
22
Linear Algebra
26
21 Familiar Linear Spaces
31
23 Differential Equations from Solutions
44
24 Characteristic Value Problems
48
The Laplace Transform
210
71 The Laplace Transform
211
72 Properties of the Laplace Transform
214
73 The Inverse Laplace Transform
225
74 Discontinous Functions and Their Transforms
230
HigherOrder Differential Equations with Variable Coefficients
240
81 CauchyEuler Differential Equations
241
82 Obtaining a Second Solution
251

FirstOrder Differential Equations
52
32 Linear Equations by Mathematica
57
33 Exact Equations
59
34 Variables Separable
69
35 Homogeneous Nonlinear Differential Equations
75
36 Bernoulli and Riccati Differential Equations Optional
79
37 Clairaut Differential Equations Optional
86
Applications of FirstOrder Equations
90
42 Linear Applications
95
43 Nonlinear Applications
117
HigherOrder Linear Differential Equations
129
51 The Fundamental Theorem
130
52 Homogeneous SecondOrder Linear Constant Coefficients
139
53 HigherOrder Constant Coefficients Homogeneous
152
54 The Method of Undetermined Coefficients
160
55 Variation of Parameters
171
Applications of SecondOrder Equations
179
62 Damped Harmonic Motion
190
63 Forced Oscillation
197
64 Simple Electronic Circuits
202
65 Two Nonlinear Examples Optional
206
83 Sums Products and Recursion Relations
253
84 Series Solutions of Differential Equations
262
85 Series Solutions About Ordinary Points
269
86 Series Solution About Regular Singular Points
277
87 Important Classical Differential Equations and Functions
293
Differential Systems Theory
297
91 Reduction to FirstOrder Systems
302
92 Theory of FirstOrder Systems
309
93 FirstOrder Constant Coefficients Systems
317
94 Repeated and Complex Roots
331
95 Nonhomogeneous Equations and BoundaryValue Problems
344
96 CauchyEuler Systems
358
Differential Systems Applications
369
102 Phase Portraits
384
103 Two Nonlinear Examples Optional
404
104 Defective Systems of FirstOrder Differential Equations Optional
410
105 Solution of Linear Systems by Laplace Transforms Optional
416
References
423
Index
426
Copyright

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Page 423 - REFERENCES 1. M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York, 1965.

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