G180 Module 07 Assignment
1.
a. Give the vertex set V.
b. Give the Edge set E.
2. Consider the graph with V = [A, B, C, X, Y, Z] and E = [AX, AY, AZ, BB, CX, CY, CZ, YY]. Without drawing a
picture of the graph:
a. List all the vertices adjacent to Y.
b. L

G180 Module 07 Assignment
1.
a. Give the vertex set V.
V = [1, 2, 3, 4, 5]
b. Give the Edge set E
E = [12, 15, 23, 24, 25, 34, 45]
2. Consider the graph with V = [A, B, C, X, Y, Z] and E = [AX, AY, AZ, BB, CX, CY, CZ, YY]. Without drawing a
picture of the

G180 Module 07 Assignment
1.
a. Give the vertex set V.
b. Give the Edge set E.
2. Consider the graph with V = [A, B, C, X, Y, Z] and E = [AX, AY, AZ, BB, CX, CY, CZ, YY]. Without drawing a
picture of the graph:
a. List all the vertices adjacent to Y.
b. L

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

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,

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

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

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

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.

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 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

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

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

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

(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

G180 Module 04 Assignment
A group of students were asked to vote on their favorite horror films. The candidate films are: Abraham
Lincoln Vampire Hunter, The Babadook, Cabin Fever, and Dead Snow (A, B, C, D for short). The following
table gives the prefer

G180 Module 06 Assignment
1. Adam, Bob and Chad are dividing an estate consisting of a house, a small farm and a painting using the
method of sealed bids. Their bids on each of the items are given in the following table:
Adam
Bob
Chad
House
$145,000
$125,

G180 Module 05 Assignment
1. A group of students were asked to vote on their favorite horror films. The candidate films are:
Abraham Lincoln Vampire Hunter, The Babadook, Cabin Fever, and Dead Snow (A, B, C, D for short). The
following table gives the pre

G180 Module 02 Assignment
Bobbie Blystone
1. A card is drawn at random out of a well-shuffled deck of 52 cards. Find the probability of the
following events.
a. Draw an Ace.
P (ACE) = 4/52
4/52= 0.0769
7.7%
b. Draw a red card
P (red card) = 26/52
26/52 =

G180 Module 01 Assignment
1. You are playing checkers with your friend. In each round you can either win or lose. List the sample
space (all possible outcomes) if you play four times in a row. Use W for win and L for lose.
W
4
3
2
1
0
L
0
1
2
3
4
2. For a

G180 Module 08 Assignment
1. Looking at the above graph, identify the number of odd vertices.
Deg(2)=3
Deg(3)=3
Deg(8)=3
Deg(9)=3
2. Looking at the above graph, identify the number of even vertices.
Deg(1)=2
Degg(4)=4
Deg(5)=2
Deg(6)=4
Deg(7)=2
Deg(10)=2

G180 Module 03 Assignment
1. Out of 120 students who took a general education math quiz, 98 passed. What percentage of the
students passed?
98/120 = 0.816
82% of students passed the quiz
2. Michael's tuition bill for last quarter was $1850. If she paid $2

G180 Module 04 Assignment
A group of students were asked to vote on their favorite horror films. The candidate films are: Abraham
Lincoln Vampire Hunter, The Babadook, Cabin Fever, and Dead Snow (A, B, C, D for short). The following
table gives the prefer

G180 Module 03 Assignment
1. Out of 420 students who took a statistic test, 310 passed. What percentage of the students passed
the test?
2. Rita's tuition bill for last semester was $4360. If she paid $6052 in tuition this semester, what was the
percentag

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