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Unformatted text preview: ‘ §OLvTI0NS UNIVERSITY OF WATERLOO
AMATII 250 MIDTERM 01, FALL 2010 VERSION A 61056 MPr‘CE Student Name (Print Legibly) (FAMILY NAME) Signature (GIVEN NAME) Student ID Number Tutorial Section (or tutorial start time) COURSE NUMBER COURSE TITLE COURSE SECTION(s) DATE OF EXAM DURATION OF EXAM NUMBER OF EXAM PAGES
INSTRUCTOR EXAM TYPE ADDITIONAL MATERIALS ALLOWED AMATH 250 Introduction to Differential Equations 001 Monday, October 18, 2010 50 minutes 5 (5 single—sided sheets) G. Mayer
Closed Book
NONE ‘ Instructions 1. Write your name, signature, ID number and tuto— rial section (or tutorial start time) on the cover.
2. Answer all questions in the spaces provided. Ask a proctor for extra blank pages if necessary. Show all your work for full marks. 3. Check that the examination has 5 single—sided
pages. 4. Your grade will be inﬂuenced by how clearly you
express your ideas, and how well you organize
your solutions. ’ Marking Scheme Question Mark Out of
cover 1
1 1 1
2 8
3 10
Total 30 “W M iV\ "‘ [JV 4’ AMATH 250  Midterm Test ()1 Fall 2010 Page 2 of 5 [5] 1. a) For each DE, indicate which type(s) it is (separable and/ or linear) and which meth»
ods apply (separation of variables, integrating factor, undetermined coefﬁcients) by placing check marks in the appropriate boxes. Types Methods .4 DE l Separable Linear Sep. Vars Int. Factor Undet. Coeﬁs. l j yl<m>=y+1 \/ v V V \/ d
ma—g—sinxzy V \/
w P
d
iii—g +siny : 3:3
dx ﬂ—xzmgf \f ' d3: MOLka “be? edzf‘ﬁ hex, Loé'tae4‘l5 rhzékwl (unlabeled) d
[6] b) Find the general solution to 8% + 4y 2 2 sinzr —— 76"436 using the method of undetermined coefﬁcients.
f
. :3. @
“ﬁx * U3 "2‘ Lk ﬁ 939” “‘15 “:5? i“ 2% geiagf «r (2ng  , as.
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a? gﬁﬁgliiée "it fists?) ﬁaSX aiiigrﬁflisiﬁx +(a ‘ix _ w‘wtx
£7 ushvga $25 V O_: "t? m ’2 ﬁts«A e? 23*l§9f‘(‘*ﬁr ‘7A;“% 5%: $44 "AMATH 250 — Midterm Test 01 Fall 2010 Page 3 0f 5 [8] 2. Consider the equation :1:— 2 2y + 6“”, whose solution is 1
y(:1:) = —ge“” + 062:”, where C’ is a constant. Provide a qualitative sketch of the solutions. Determine where
solutions of the DE are increasing/decreasing, and ﬁnd any equilibrium and exceptional
solutions. Also discuss the behaviour of the solutions as :1: tends to positive inﬁnity and
negative inﬁnity. 0 ' " “x \‘5' 1e
f “at z: L’NS
%M E {war a cagth lkwtis CW; villa wait}: :5 ,. AMATH 250 — Midterm Test 01 Fall 2010 Page 4 of 5 3. The temperature of a hot cup of coffee can be modeled with Newton’s law of cooling,
described by the initial value problem if; = —k(T — Ta), T(0) = Tea which has the solution
m) = Ta + (T0 — med“. The parameter k is an unknown positive constant and T is the temperature of the coffee at time t.
[1] a) State what Ta represents.
[2] ose that the coffee, initially at 100°C, is placed in a room whose temperature is / 20° . Find an expression in terms of k for the time at which the temperature of the co ee reaches 50°C. k [g go “=40 <1~C§ee~wj%¢" .
net: as» 2% WW) Cog u: w WD] [1:] JC 1“”{4 [7] c) Suppose that when t = 0, the coffee is 100°C, and a small object with initial temper—
ature 130°C is placed inside the coffee and starts to cool down. Use Newton’s law of
cooling to ﬁnd an expression formthe temperature U (t) of the small object. Assume that timx w z
I \N (7’3 5;]: (E W tiger; ﬁMBiEN: T on, "T: 0 temperature of the room/ is 10°C V Co the small object has thei‘same theWroperties of the coffee so that they have
the same It WNW o the object is small enough so that it does not affect the temperature of the coffee
0 the object is completely surrounded by coffee ¥ ,_ , ﬁst WWW
8 LL] 3 \S ’ :ﬂk (U‘ﬂt‘igéqeé WT. fACWK’
Q] Mi 3, km a {oig + one’l‘i’ § (eftmlilf‘i 4344,: UNBQ’L Co§?F. 2 W e (,9, “gt 4 a We].
’ ‘ a i
E L}F 13%th Rf) ‘ {(0 Hint: Newton’s law of cooling is also valid when Ta is not constant. U; a 82C?“ “is”? ’55“
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