20.9 Tridiagonalization and QR-Factorization 888
Chapter 20 Review Questions and Problems 896
Summary of Chapter 20 898
CHAPTER 21
Numerics for ODEs and PDEs 900
21.1 Methods for First-Order ODEs 901
21.2 Multistep Methods 911
21.3 Methods for Systems and

solution. Substituting and in the last equation gives Hence
Hence the amount of salt in the tank at time t is
(5)
This function shows an exponential approach to the limit 5000 lb; see Fig. 11. Can you explain physically that
should increase with time? Tha

The presentation in this book is adaptable to various degrees of use of software,
Computer Algebra Systems (CASs), or programmable graphic calculators, ranging
from no use, very little use, medium use, to intensive use of such technology. The choice
of ho

solution of the IVP.
9.
10.
11.
12.
13.
14.
15. Find two constant solutions of the ODE in Prob. 13 by
inspection.
16. Singular solution. An ODE may sometimes have an
additional solution that cannot be obtained from the
general solution and is then called

Direction Fields, Eulers Method
A first-order ODE
(1)
has a simple geometric interpretation. From calculus you know that the derivative of
is the slope of . Hence a solution curve of (1) that passes through a point
must have, at that point, the slope equa

(1)
by purely algebraic manipulations. Then we can integrate on both sides with respect to x,
obtaining
(2)
On the left we can switch to y as the variable of integration. By calculus, , so that
(3)
If f and g are continuous functions, the integrals in (3)

Chapter 14 Review Questions and Problems 668
Summary of Chapter 14 669
CHAPTER 15
Power Series, Taylor Series 671
15.1 Sequences, Series, Convergence Tests 671
15.2 Power Series 680
15.3 Functions Given by Power Series 685
15.4 Taylor and Maclaurin Series

Introduction to Numerics rewritten for greater clarity and better presentation; new
Example 1 on how to round a number. Sec. 19.3 on interpolation shortened by
removing the less important central difference formula and giving a reference instead.
Large

CHAPTER 1 First-Order ODEs
CHAPTER 2 Second-Order Linear ODEs
CHAPTER 3 Higher Order Linear ODEs
CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods
CHAPTER 5 Series Solutions of ODEs. Special Functions
CHAPTER 6 Laplace Transforms
Many physical l

or typing on your computer, but first without the aid of a CAS). In doing so, you will
gain an important conceptual understanding and feel for the basic terms, such as ODEs,
direction field, and initial value problem. If you wish, you can use your Compute

solution).
Geometrically, the general solution of an ODE is a family of infinitely many solution
curves, one for each value of the constant c. If we choose a specific c (e.g., or 0
or ) we obtain what is called a particular solution of the ODE. A particul

CHAPTER 2
Second-Order Linear ODEs 46
2.1 Homogeneous Linear ODEs of Second Order 46
2.2 Homogeneous Linear ODEs with Constant Coefficients 53
2.3 Differential Operators. Optional 60
2.4 Modeling of Free Oscillations of a MassSpring System 62
2.5 EulerCau

19.
Sol.
20.
Sol. y _ 1>(1 _ x)5
yr _ _5x4y2, y(0) _ 1, h _ 0.2
y _ x _ tanh x
yr _ (y _ x)2, y(0) _ 0, h _ 0.1
yr _ y, y(0) _ 1, h _ 0.01
yr _ y, y(0) _ 1, h _ 0.1
E X A M P L E 2 Separable ODE
The ODE is separable; we obtain
E X A M P L E 3 Initial Valu

CHAPTER 4
Systems of ODEs. Phase Plane. Qualitative Methods 124
4.0 For Reference: Basics of Matrices and Vectors 124
4.1 Systems of ODEs as Models in Engineering Applications 130
4.2 Basic Theory of Systems of ODEs. Wronskian 137
4.3 Constant-Coefficient

(SturmLiouville Problems) and Sec. 5.8 (Orthogonal Eigenfunction Expansions) and
moved material into Chap. 11 (see Major Changes above).
New equivalent definition of basis (Sec. 7.4).
In Sec. 7.9, completely new part on composition of linear transformat

10.6 Surface Integrals 443
10.7 Triple Integrals. Divergence Theorem of Gauss 452
10.8 Further Applications of the Divergence Theorem 458
10.9 Stokess Theorem 463
Chapter 10 Review Questions and Problems 469
Summary of Chapter 10 470
P A R T C Fourier Ana

E X A M P L E 4 Initial Value Problem
Solve the initial value problem
Solution.
The general solution is ; see Example 3. From this solution and the initial condition
we obtain Hence the initial value problem has the solution . This is a
particular solutio

h
C
L
E
R
y
t
y
1
C
y = ky1y2 ly2
y = ay1 by1 y1 2
2
(Sec. 2.8)
y+ w0
2
y = cos wt, w0 w
Fig. 2. Some applications of differential equations
are ordinary differential equations (ODEs). Here, as in calculus, denotes ,
etc. The term ordinary distinguishes t

yr _ 1 _ y2, (14
p, 1)
(x, y)
of equal inclination) of an autonomous ODE look like?
Give reason.
1215 MOTIONS
Model the motion of a body B on a straight line with
velocity as given, being the distance of B from a point
at time t. Graph a direction field o

from the direction field.
(d) Graph the direction field of and some
solutions of your choice. How do they behave? Why
do they decrease for y _ 0?
yr _ _12
y
yr _ _x>y
x2 _ 9y2 _ c (y _ 0).
_5 _ x _ 2, _1 _ y _ 5.
1720 EULERS METHOD
This is the simplest me

they will be considered in Chap. 12.
An ODE is said to be of order n if the nth derivative of the unknown function y is the
highest derivative of y in the equation. The concept of order gives a useful classification
into ODEs of first order, second order,

Here, open interval means that the endpoints a and b are not regarded as
points belonging to the interval. Also, includes infinite intervals
(the real line) as special cases.
E X A M P L E 1 Verification of Solution
Verify that (c an arbitrary constant) i

4 0.8 0.274 0.426 0.152
5 1.0 0.488 0.718 0.230
xn yn y(xn)
18 DIRECTION FIELDS, SOLUTION CURVES
Graph a direction field (by a CAS or by hand). In the field
graph several solution curves by hand, particularly those
passing through the given points .
1.
2.

These problems will give you a first impression of modeling.
Many more problems on modeling follow throughout this
chapter.
17. Half-life. The half-life measures exponential decay.
It is the time in which half of the given amount of
radioactive substance