Homework #2 (5286301)
Due:
Sun Jan 26 2014 11:59 PM EST
Question
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9 10 11 12 13 14 15
Instructions
For the Direction Field problems, you are encouraged to use Maple or some other
graphical tool to explore the direction fields, but for the t
Math 2003 * Assignment 8 * Existence and Uniqueness
In this assignment, by Theorem we mean the theorem about the existence and
uniqueness of solutions for ﬁrst order ODEs (Theorem 4.1 in the notes)
1. Write down the statement of the Theorem.
vaeiolm W N
MATHEMATICS 2C03
.
DURATION: 1 HOUR
McMASTER UNIVERSITY TERM TEST 1
Dr. G. S. K. Wolkowicz
February 12, 2015
THIS TEST PAPER INCLUDES 8 PAGES and 6 QUESTIONS. IT IS POSSIBLE TO OBTAIN A
TOTAL OF 50 MARKS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF
Mathematics 2C03 Test 2
Name:
Student ID:
MULTIPLE CHOICE
There are 3 multiple choice questions. All questions in this section have the same value. A
correct answer scores 4 and an incorrect answer scores zero. Record your answer by circling ONE
and ONLY
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=
ES
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/HNF MB M ?I A FS ' L ? RC@
MATHEMATICS 2C03
.
DURATION: 1 HOUR
McMASTER UNIVERSITY PRACTICE TERM TEST 2
THIS TEST PAPER INCLUDES 10 PAGES AND 6 QUESTIONS. IT IS POSSIBLE TO OBTAIN
A TOTAL OF 40 MARKS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF
THE PAPER IS COMPLETE. BRING A
Mathematics 2C03 Test 1
Name:
Student ID:
MULTIPLE CHOICE
There are 3 multiple choice questions. All questions in this section have the same value. A
correct answer scores 4 and an incorrect answer scores zero. Record your answer by circling ONE
and ONLY
Phase Lines
1. Denition
.
Since the direction eld for an autonomous DE, y = f (y), is constant
on horizontal lines, its essential content can be conveyed more efciently
using the following recipe:
1. Draw the y-axis as a vertical line and mark on it the e
MATHEMATICS 2C03
.
DURATION: 1 HOUR
McMASTER UNIVERSITY TERM TEST 1
Dr. G. S. K. Wolkowicz
February 12, 2015
THIS TEST PAPER INCLUDES 8 PAGES and 6 QUESTIONS. IT IS POSSIBLE TO OBTAIN A
TOTAL OF 50 MARKS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF
MATHEMATICS 2C03
.
DURATION: 1 HOUR
McMASTER UNIVERSITY PRACTICE TERM TEST 2
THIS TEST PAPER INCLUDES 10 PAGES AND 6 QUESTIONS. IT IS POSSIBLE TO OBTAIN
A TOTAL OF 40 MARKS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF
THE PAPER IS COMPLETE. BRING A
Bessels equation with = 1/2 - the 2nd solution
Chai Molina
July 23, 2014
In class, we discussed Bessels equation,
x2 y + xy + (x2 2 )y = 0,
(1)
with = 1/2. We wanted to nd a fundamental set of solutions about the
regular singular point x0 = 0. As we saw,
Mathematics 2C03 Test 1
Name:
Student ID:
MULTIPLE CHOICE
There are 3 multiple choice questions. All questions in this section have the same value. A
correct answer scores 4 and an incorrect answer scores zero. Record your answer by circling ONE
and ONLY
Math 2C03 Summer 2014:
Test 1 Comments:
Please read the posted solutions carefully.
Question 2:
To find the critical harvesting rate rK/4 as stated in the solutions, we want to find when all
solutions decrease to y=0. For this, we look for equilibrium sol
Directional Fields
Set y equal to zero and look at how the derivative behaves along the xaxis.,
Do the same for the y-axis by setting x equal to 0.
Consider the curve in the plane defined by setting y0 = 0.
This should correspond to the points in the pict
Hrt)
Ix 3:/*?31*(!2-Y2) =
v :5 cfw_Ac onlv
mu.
w- :waJ 26:) =1 clam-V
TLv ASS-c.'a4LC,- 74! .$ by
xy" x.P(o)y'4 may: a
x17 1.x .|-7 * -"-" O so"! 0 36336: (.7!- in
'1qu 4 h Gauge
91" _ '2'." 6- .
well Iced MK .v='/z ,m.s+l,.
i 7 9 D B 4.34:0
a) ' f ' )ag
Mam/k ZCOC) ~ Aggrammwg #4
?Oww SUM-'5 /fle.luLJcb" 94(- QfoLU/', Earl-13
11) Eat X13K\3'+\3:0 \5 0-. CE 91:"-
Am 91: o:\(Al) q H: whiz"amt : (x431.
("n+5 = 4 MUH:[,'.(.'~;:'L
:) aim- qx+ szlwx,
(H'LH quaxlwx)
In W tg aK Pmrhwynr 3.3; ,5, A a L. a.
2 n e"
MATHEMATICS 2C03
.
DURATION: 60 MINUTES
McMASTER UNIVERSITY TERM TEST 2 v1
Dr. G. S. K. Wolkowicz
March 19, 2015
THIS TEST PAPER INCLUDES 10 PAGES, 6 QUESTIONS. IT IS POSSIBLE TO OBTAIN A
TOTAL OF 50 MARKS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY
Math 2003 * Assignment 10 * Exact ODES
1. Determine whether each ODE is exact or not.
(a) de + [23; sin(a:y) + 33312 cos(33y)]dy = 0
M N
2 “a \ MYXQNK
ME: 005003) Hi xﬂgmmsy % ‘ 1E EXACT
N K 2: 1&2003LX‘33'4—kt‘2-CQ‘5LKK63 "‘ X‘ﬁ 3‘“
$2 I: a: 2 3.
(b)
Math 2003 * Assignment 9 * First-Order Linear ODEs
1. Recall that a general form of a ﬁrst-order linear ODE is y’ + P(:I:)y = Q($), where
P(a:) and 62(3) are ﬁnctiens of a: only.
In each case, state whether a given ODE is separable, linear, or neither
Math 2003 * Assignment 5 * Slope ﬁelds
1. Consider a ﬁrst—order IVP y’ : G(w, y), Mam) 2 yo.
(a) What is the purpose of a slope ﬁeld? How is it created?
* To O‘QWW 0AA WMKCMOA'K golul‘x‘mm OQM 1W lﬁh-
wmin—QNMK 06 W Whack (wave) %M\ l
.K Cowul-k M Q1911;
Math 2003 * Assignment 6 * Qualitative Analysis
1. Consider the ODE y’ = 932(y ~ 1).
(a) Find all values of for which the solution y(:.v) is increasing.
win.“ $470 (C‘e~,\g70,+€«m 317/0 «4? kg immenswa
Cnme about 20 Kéoome 332$ £10)
(b) Find all values of
Math 2003 * Assignment 4 * Solving ODES
1. Find all values of a for which :1: I eat is a solution of the ODE 333’ —l— 173 2 0.
a}: l ai: ‘
K26 "97"1‘1‘3 673+ GK = 730er MS? 6ij Z0
-~—+ QG&K%&+$1):O
W“
30% H “=0 «=9 olz—ﬁ’l?)
1’?-
HERE
._9 X: e_ {0: q So
Math 2003 * Assignment 1 * Review of Differentiation (ODEs)
1. Show that y : C's—333, where C is any constant, is a, solution of the ODE y’ + 3:323; : 0.
2. Show that a: : + C's—3W2, where C is a, real number, is a solution of the ODE
93’ + 371$ : t.
Numerical Reasoning
Free Test 1
Solutions Booklet
Instructions
This numerical reasoning test comprises 21 questions, and you will have 21 minutes in
which to correctly answer as many as you can. Calculators are permitted for this test, and it is
recommend
Section 2 Dierential Equations: Basics
2
21
Dierential Equations: Basics
Derivatives: Notation, Symbols and Denitions
We will be using dierent sets of notations for the derivatives. In case of a
function of one variable, we will use the common convention
Section 1 Introduction: Dierential Equations
1
1
Introduction: Dierential Equations
To study real-world problems, we often use functions to describe the quantity
(quantities) involved. For instance, a function P (t) can be used to track the
number of bact
MATHEMATICS 2C3
TEST 1
Day Class
Duration of Examination: 50 minutes
McMaster University, 8 February 2016
Dr. M. Lovric
FIRST NAME (please print):
FAMILY NAME (please print):
Student No.:
THIS TEST HAS 7 PAGES AND 7 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSU
Math 2C3 * Course Outline
file:/courses pages 2016:17/2C3/2C3mathoutline.html
IMPORTANT MESSAGE: COUSE OUTLINE IS TENTATIVE
The instructor and university reserve the right to modify elements of the course during the term. The
university may change the dat