9780387978949

Differential Equations and Their Applications : An Introduction to Applied Mathematics

by
  • ISBN13:

    9780387978949

  • ISBN10:

    0387978941

  • Format: Hardcover
  • Copyright: 1/1/1993
  • Publisher: Springer Verlag

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Summary

Used in undergraduate classrooms across the country, this book is a clearly written, rigorous introduction to differential equations and their applications. Fully understandable to students who have had one year of calculus, this book differentiates itself from other differential equations texts through its engaging application of the subject matter to interesting scenarios. This fourth edition incorporates earlier introductory material on bifurcation theory and adds a new chapter on Sturm-Liouville boundary value problems. Computer programs in C, Pascal, and Fortran are presented throughout the text to show the read how to apply differential equations towards quantitative problems.

Table of Contents

First-order differential equations
1(126)
Introduction
1(1)
First-order linear differential equations
2(9)
The Van Meegeren art forgeries
11(9)
Separable equations
20(6)
Population models
26(13)
The spread of technological innovations
39(7)
An atomic waste disposal problem
46(6)
The dynamics of tumor growth, mixing problems, and orthogonal trajectories
52(6)
Exact equations, and why we cannot solve very many differential equations
58(9)
The existence-uniqueness theorem; Picard iteration
67(14)
Finding roots of equations by iteration
81(10)
Newton's method
87(4)
Difference equations, and how to compute the interest due on your student loans
91(5)
Numerical approximations; Euler's method
96(11)
Error analysis for Euler's method
100(7)
The three term Taylor series method
107(2)
An improved Euler method
109(3)
The Runge-Kutta method
112(4)
What to do in practice
116(11)
Second-order linear differential equations
127(137)
Algebraic properties of solutions
127(11)
Linear equations with constant coefficients
138(13)
Complex roots
141(4)
Equal roots; reduction of order
145(6)
The nonhomogeneous equation
151(2)
The method of variation of parameters
153(4)
The method of judicious guessing
157(8)
Mechanical vibrations
165(13)
The Tacoma Bridge disaster
173(2)
Electrical networks
175(3)
A model for the detection of diabetes
178(7)
Series solutions
185(40)
Singular points, Euler equations
198(5)
Regular singular points, the method of Frobenius
203(16)
Equal roots, and roots differing by an integer
219(6)
The method of Laplace transforms
225(8)
Some useful properties of Laplace transforms
233(5)
Differential equations with discontinuous right-hand sides
238(5)
The Dirac delta function
243(8)
The convolution integral
251(6)
The method of elimination for systems
257(2)
Higher-order equations
259(5)
Systems of differential equations
264(108)
Algebraic properties of solutions of linear systems
264(9)
Vector spaces
273(6)
Dimension of a vector space
279(12)
Applications of linear algebra to differential equations
291(6)
The theory of determinants
297(13)
Solutions of simultaneous linear equations
310(10)
Linear transformations
320(13)
The eigenvalue-eigenvector method of finding solutions
333(8)
Complex roots
341(4)
Equal roots
345(10)
Fundamental matrix solutions; eAt
355(5)
The nonhomogeneous equation; variation of parameters
360(8)
Solving systems by Laplace transforms
368(4)
Qualitative theory of differential equations
372(104)
Introduction
372(6)
Stability of linear systems
378(7)
Stability of equilibrium solutions
385(9)
The phase-plane
394(4)
Mathematical theories of war
398(16)
L.F. Richardson's theory of conflict
398(7)
Lanchester's combat models and the battle of Iwo Jima
405(9)
Qualitative properties of orbits
414(4)
Phase portraits of linear systems
418(10)
Long time behavior of solutions; the Poincare-Bendixson Theorem
428(9)
Introduction to bifurcation theory
437(6)
Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I
443(8)
The principle of competitive exclusion in population biology
451(7)
The Threshold Theorem of epidemiology
458(7)
A model for the spread of gonorrhea
465(11)
Separation of variables and Fourier series
476(38)
Two point boundary-value problems
476(5)
Introduction to partial differential equations
481(2)
The heat equation; separation of variables
483(4)
Fourier series
487(6)
Even and odd functions
493(5)
Return to the heat equation
498(5)
The wave equation
503(5)
Laplace's equation
508(6)
Sturm---Liouville boundary value problems
514(31)
Introduction
514(1)
Inner product spaces
515(11)
Orthogonal bases, Hermitian operators
526(7)
Sturm-Liouville theory
533(12)
Appendix A Some simple facts concerning functions of several variables 545(2)
Appendix B Sequences and series 547(2)
Appendix C C Programs 549(8)
Answers to odd-numbered exercises 557(18)
Index 575

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