# Differential Equations: An Introduction with Mathematica®

Springer Science & Business Media, Aug 3, 2004 - Mathematics - 434 pages
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

### Popular passages

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