Differential Equations with Boundary Value Problems

by Polking, John; Boggess, Albert; Arnold, David

• ISBN13: 9780130911063
• ISBN10: 0130911062
• Edition: 1st
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Prentice Hall

Summary

Ideal for one or two term first courses in differential equations for engineering, math, biology, and finance majors. This text strikes a balance between the traditional and the modern. It combines the traditional material with a modern systems emphasis. It offers flexibility of use that will allow faculty at a variety of institutions to use the book with extensive problem sets.

Table of Contents

Preface	ix
Introduction to Differential Equations	1	(17)
Differential Equation Models	2	(5)
The Derivative	7	(3)
Integration	10	(8)
First-Order Equations	18	(105)
Differential Equations and Solutions	18	(13)
Solutions to Separable Equations	31	(13)
Models of Motion	44	(10)
Linear Equations	54	(10)
Mixing Problems	64	(9)
Exact Differential Equations	73	(17)
Existence and Uniqueness of Solutions	90	(12)
Dependence of Solutions on Initial Conditions	102	(5)
Autonomous Equations and Stability	107	(16)
Project 2.10 The Daredevil Skydiver	120	(3)
Modeling and Applications	123	(40)
Modeling Population Growth	124	(14)
Models and the Real World	138	(5)
Personal Finance	143	(9)
Electrical Circuits	152	(11)
Project 3.5 The Spruce Budworm	158	(3)
Project 3.6 Social Security, Now or Later	161	(2)
Second-Order Equations	163	(68)
Definitions and Examples	163	(11)
Second-Order Equations and Systems	174	(5)
Linear, Homogeneous Equations with Constant Coefficients	179	(11)
Harmonic Motion	190	(9)
Inhomogeneous Equations; the Method of Undetermined Coefficients	199	(10)
Variation of Parameters	209	(6)
Forced Harmonic Motion	215	(16)
Project 4.8 Nonlinear Oscillators	228	(3)
The Laplace Transform	231	(70)
The Definition of the Laplace Transform	232	(9)
Basic Properties of the Laplace Transform	241	(7)
The Inverse Laplace Transform	248	(8)
Using the Laplace Transform to Solve Differential Equations	256	(10)
Discontinuous Forcing Terms	266	(14)
The Delta Function	280	(7)
Convolutions	287	(11)
Summary	298	(3)
Project 5.9 Forced Harmonic Oscillators	299	(2)
Numerical Methods	301	(34)
Euler's Method	302	(12)
Runge-Kutta Methods	314	(8)
Numerical Error Comparisons	322	(5)
Practical Use of Solvers	327	(5)
A Cautionary Tale	332	(3)
Project 6.6 Numerical Error Comparison	334	(1)
Matrix Algebra	335	(61)
Vectors and Matrices	335	(13)
The Geometry of Systems of Linear Equations	348	(5)
Solving Systems of Equations	353	(9)
Properties of Solution Sets	362	(9)
Subspaces	371	(13)
Determinants	384	(12)
An Introduction to Systems	396	(48)
Definitions and Examples	396	(9)
Geometric Interpretation of Solutions	405	(12)
Qualitative Analysis	417	(8)
Linear Systems	425	(19)
Project 8.5 Long-Term Behavior of Solutions	441	(3)
Linear Systems with Constant Coefficents	444	(101)
Overview of the Technique	444	(8)
Planar Systems	452	(14)
Phase Plane Portraits	466	(18)
Higher Dimensional Systems	484	(8)
The Exponential of a Matrix	492	(18)
Qualitative Analysis of Lienar Systems	510	(5)
Higher-Order Linear Equations	515	(13)
Inhomogeneous Linear Systems	528	(17)
Project 9.9 Phase Plane Portraits	538	(1)
Project 9.10 Oscillations of Linear Molecules	539	(6)
Nonlinear Systems	545	(92)
The Linearization of a Nonlinear System	545	(14)
Long-Term Behavior of Solutions	559	(7)
Invariant Sets and the Use of Nullclines	566	(8)
Long-Term Behavior of Solutions to Planar Systems	574	(12)
Conserved Quantities	586	(6)
Nonlinear Mechanics	592	(17)
The Method of Lyapunov	609	(10)
Predator-Prey Systems	619	(18)
Project 10.9 Human Immune Response to Infectious Disease	631	(3)
Project 10.10 Analysis of Competing Species	634	(3)
Series Solutions to Differential Equations	637	(75)
Review of Power Series	638	(12)
Series Solutions Near Ordinary Points	650	(12)
Legendre's Equation	662	(6)
Types of Singular Points---Euler's Equation	668	(9)
Series Solutions Near Regular Singular Points	677	(13)
Solutions in the Exceptional Cases	690	(11)
Bessel's Equation and Bessel Functions	701	(11)
Fourier Series	712	(38)
Computation of Fourier Series	713	(11)
Convergence of Fourier Series	724	(9)
Fourier Cosine and Sine Series	733	(5)
The Complex Form of a Fourier Series	738	(3)
The Discrete Fourier Transform and the FFT	741	(9)
Partial Differential Equations	750
Derivation of the Heat Equation	750
Separation of Variables for the Heat Equation	756
The Wave Equation	768
Laplace's Equation	778
Laplace's Equation on a Disk	785
Sturm Liouville Problems	791
Orthogonality and Generalized Fourier Series	801
Temperature in a Ball---Legendre Polynomials	810
Time Dependent PDEs in Higher Dimension	814
Domains with Circular Symmetry---Bessel Functions	821
Answers to Odd-Numbered Problems	A-1
Index	I-1