Differential Equations With Applications,9780486411545

Differential Equations With Applications

by ;
Edition: Unabridged
ISBN13: 9780486411545
ISBN10: 0486411540
Format: Paperback
Pub. Date: 2010-08-19
Publisher(s): Dover Publications
List Price: $19.15

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Summary

Coherent, balanced introductory text focuses on initial- and boundary-value problems, general properties of linear equations, and the differences between linear and nonlinear systems. Includes large number of illustrative examples worked out in detail and extensive sets of problems. Answers or hints to most problems appear at end.

Table of Contents

Preface vii
Basic Concepts
1(35)
Introduction
1(1)
Classifications and Examples of Differential Equations
1(3)
The Motion of a Particle
4(5)
The Solution of a Differential Equation
9(6)
Initial- and Boundary-value Problems
15(5)
The Differential Equation y' = f(x)
20(3)
Integrals as Functions of Parameters
23(4)
Elementary and Non-elementary Functions
27(1)
The Gamma Function
28(8)
Appendix
31(5)
Special Methods for First-order Equations
36(31)
Introduction
36(1)
Separation of Variables
36(4)
The First-order Linear Differential Equation
40(7)
Exact Differential Equations
47(5)
Integrating Factors
52(5)
Use of Substitutions
57(5)
Second-order Equations Reducible to First-order
62(2)
Summary
64(3)
Applications of First-order Equations
67(28)
Introduction
67(1)
Falling Bodies with Air Resistance
67(4)
Motion on a Given Curve
71(4)
Linear Motion with Variable Mass
75(2)
Newton's Law of Cooling
77(3)
Dilution Problems
80(1)
Chemical Reactions
81(2)
Population Growth
83(2)
A Simple Electrical Circuit
85(6)
Families of Curves and Orthogonal Trajectories
91(4)
Existence and Uniqueness and Methods of Approximation
95(24)
Introduction
95(1)
The Direction Field
95(5)
Existence and Uniqueness of Solutions
100(3)
The Picard Method
103(4)
The Cauchy-Euler Method
107(2)
Taylor Series
109(3)
Existence and Uniqueness Theorems for Systems of Equations and Higher-order Equations
112(2)
Existence and Uniqueness Theorems for Linear Equations
114(5)
Appendix: Proof of existence and uniqueness theorems
115(4)
Linear Differential Equations
119(37)
Introduction
119(1)
Fundamental Theory of Second-order Linear Equations
119(7)
Complex-valued Solutions
126(3)
Homogeneous Linear Equations with Constant Coefficients
129(6)
Undetermined Coefficients
135(6)
Variation of Parameters
141(3)
Euler's Equation
144(3)
Formulas of Lagrange and Abel
147(2)
Linear Equations of the nth Order
149(7)
Applications of Second-order Linear Differential Equations
156(29)
Introduction
156(1)
Free Vibrations
157(9)
Forced Vibrations
166(6)
Electrical Circuits
172(8)
The Equations of Planetary Motion
180(5)
Linear Differential Equations with Variable Coefficients
185(39)
Introduction
185(1)
Solution by Power Series
186(10)
Solution near a Singular Point
196(8)
Bessel's Equation
204(12)
Hypergeometric Equation
216(2)
Legendre's Equation
218(6)
Systems of Linear Differential Equations
224(43)
Introduction
224(1)
Some Illustrative Examples
224(5)
A Two-degree-of-freedom Vibration
229(4)
Vectors and Matrices
233(6)
Theory of Systems of Linear Differential Equations
239(8)
Homogeneous Linear Systems with Constant Coefficients
247(4)
Solution by Matrix Methods
251(16)
The Laplace Transform
267(41)
Introduction
267(1)
Improper Integrals
267(1)
The Laplace Transform
268(5)
Properties of the Laplace Transform
273(6)
Solution of Linear Equations with Constant Coefficients
279(6)
Product of Transform Functions; Convolutions
285(4)
Discontinuous Functions
289(4)
Linear Systems Analysis
293(15)
Appendix
303(5)
Nonlinear Differential Equations
308(27)
Introduction
308(1)
The Pendulum
309(6)
Singularities and the Phase Plane
315(9)
Van der Pol's Equation
324(3)
Piecewise Linear System
327(3)
Liapunov's Second Method
330(5)
Linear Difference Equations
335(38)
Introduction
335(1)
First-order Linear Difference Equations
336(4)
Second-order Linear Difference Equations
340(4)
Homogeneous Linear Difference Equations with Constant Coefficients
344(4)
The Nonhomogeneous Equation
348(7)
The Vector Space EN
355(8)
A Boundary-value Problem
363(4)
N Beads on a Tightly Stretched String
367(6)
Numerical Methods
373(26)
Introduction
373(1)
The Euler Method
374(3)
Error Analysis
377(5)
Parasitic Solutions and Stability
382(3)
A Second-order Predictor-Corrector Method
385(4)
Fourth-order Predictor-Corrector Methods
389(4)
Starting Methods and Runge-Kutta Methods
393(3)
Higher-order Equations and Systems of Equations
396(3)
Boundary-value Problems
399(45)
Introduction
399(1)
Homogeneous Boundary-value Problems
399(5)
Eigenvalue Problems
404(5)
Orthogonal Functions
409(5)
Generalized Fourier Series
414(6)
Weight Functions
420(1)
The Sturm-Liouville Problem
421(3)
Theorems on Eigenvalues and Eigenfunctions
424(2)
Ordinary Fourier Series
426(7)
Fourier-Bessel Series
433(2)
Fourier-Legendre Series
435(3)
Nonhomogeneous Boundary-value Problems
438(6)
Partial Differential Equations of Mathematical Physics
444(32)
Introduction
444(1)
The Vibrating String
445(14)
Heat Conduction
459(6)
Laplace's Equation
465(6)
Theory of Second-order Equations
471(5)
Further Applications of Partial Differential Equations
476(17)
Introduction
476(1)
Laplace's Equation in Three Dimensions
476(8)
Temperature in an Infinite Cylinder
484(2)
Vibrating Membranes
486(7)
Appendix A Infinite Series 493(5)
Appendix B Functions of a Complex Variable 498(5)
References 503(2)
Answers and Hints 505(32)
Index 537

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