Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Elementary
Differential
Equations and
Boundary Value
Problems
S E V E N T H E D I T I O N
Elementary Differential
Equations and Boundary
Value Problems
William E. Boyce
Edward P. Hamil...
ASSOCIATE EDITOR Mary Johenk
MARKETING MANAGER Julie Z. Lindstrom
PRODUCTION EDITOR Ken Santor
COVER DESIGN Michael Jung
I...
To Elsa and Maureen
To Siobhan, James, Richard, Jr., Carolyn, and Ann
And to the next generation:
Charles, Aidan, Stephani...
The Authors
William E. Boyce received his B.A. degree in Mathematics from Rhodes College,
and his M.S. and Ph.D. degrees i...
P R E F A C E
This edition, like its predecessors, is written from the viewpoint of the applied mathe-
matician, whose int...
viii Preface
Mathematical Modeling
The main reason for solving many differential equations is to try to learn something
ab...
Preface ix
combinations are also possible and have been used effectively with earlier editions of
this book.
With respect ...
x Preface
Coombes, B. R. Hunt, R. L. Lipsman, J. E. Osborn, and G. J. Stuck, all at the University
of Maryland, provide de...
Preface xi
5. An example illustrating the instabilities that can be encountered when dealing
with stiff equations has been...
A C K N O W L E D G M E N T S
It is a pleasure to offer my grateful appreciation to the many people who have generously
as...
Acknowledgments xiii
Most important, I thank my wife Elsa for many hours spent proofreading and check-
ing details, for ra...
C O N T E N T S
Preface vii
Chapter 1 Introduction 1
1.1 Some Basic Mathematical Models; Direction Fields 1
1.2 Solutions ...
Contents xv
4.3 The Method of Undetermined Coefficients 222
4.4 The Method of Variation of Parameters 226
Chapter 5 Series ...
xvi Contents
8.3 The Runge–Kutta Method 435
8.4 Multistep Methods 439
8.5 More on Errors; Stability 445
8.6 Systems of Fir...
C H A P T E R
1
Introduction
In this chapter we try in several different ways to give perspective to your study of
differe...
2 Chapter 1. Introduction
circuits, the dissipation of heat in solid objects, the propagation and detection of
seismic wav...
1.1 Some Basic Mathematical Models; Direction Fields 3
and Eq. (2) then becomes
m
dv
dt
= mg − γ v. (4)
Equation (4) is a ...
4 Chapter 1. Introduction
2 4 t6 8 10
48
44
40
52
60
56
υ
FIGURE 1.1.2 A direction field for Eq. (5).
no graphs of solution...
1.1 Some Basic Mathematical Models; Direction Fields 5
The approach illustrated in Example 2 can be applied equally well t...
6 Chapter 1. Introduction
into the model, we must add another term to the differential equation (7), so that it
becomes
dp...
1.1 Some Basic Mathematical Models; Direction Fields 7
where the growth rate r and the predation rate k are unspecified. So...
8 Chapter 1. Introduction
PROBLEMS In each of Problems 1 through 6 draw a direction field for the given differential equati...
1.2 Solutions of Some Differential Equations 9
In each of Problems 19 through 26 draw a direction field for the given diffe...
10 Chapter 1. Introduction
Since, by the chain rule, the left side of Eq. (6) is the derivative of ln |p − 900| with
respe...
1.2 Solutions of Some Differential Equations 11
might have. This is typical of what happens when you solve a differential ...
12 Chapter 1. Introduction
a particular value of c, and is the graph of the solution corresponding to that value of c.
Sat...
1.2 Solutions of Some Differential Equations 13
By integrating both sides we obtain
ln |v − 49| = −
t
5
+ C, (24)
and then...
14 Chapter 1. Introduction
where c is an arbitrary constant of integration. The object starts to fall when t = 0,
so we kn...
1.2 Solutions of Some Differential Equations 15
᭤ 2. Follow the instructions for Problem 1 for the following initial value...
16 Chapter 1. Introduction
10. A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportion...
1.3 Classification of Differential Equations 17
system can take water from the pool, remove the dye, and return the water t...
18 Chapter 1. Introduction
Systems of Differential Equations. Another classification of differential equations
depends on t...
1.3 Classification of Differential Equations 19
Linear and Nonlinear Equations. A crucial classification of differential equ...
20 Chapter 1. Introduction
by linear equations; to study these phenomena it is essential to deal with nonlinear
equations....
1.3 Classification of Differential Equations 21
has solutions. However, this is not a purely mathematical concern, for at l...
22 Chapter 1. Introduction
illuminating and helpful in understanding and interpreting the solution of a differential
equat...
1.4 Historical Remarks 23
10. y + 4y + 3y = t; y1(t) = t/3, y2(t) = e−t
+ t/3
11. 2t2
y + 3ty − y = 0, t > 0; y1(t) = t1/2...
24 Chapter 1. Introduction
the subject, and identify the most prominent early contributors. Other historical infor-
mation...
1.4 Historical Remarks 25
Problem 33 of Section 2.3). The brachistochrone problem was also solved by Leibniz
and Newton in...
26 Chapter 1. Introduction
The Laplace transform (Chapter 6) is also named for him although its usefulness in
solving diff...
1.4 Historical Remarks 27
For further reading in the history of mathematics see books such as those listed below:
Boyer, C...
C H A P T E R
2
First Order
Differential
Equations
This chapter deals with differential equations of first order,
dy
dt
= f...
30 Chapter 2. First Order Differential Equations
type of first order linear equation in which the coefficients are constants...
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
calculas7
Upcoming SlideShare
Loading in …5
×

calculas7

1,329 views

Published on

Published in: Education, Technology