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Unformatted text preview: UNIVERSITY OF WATERLOO TERM TEST #1
SPRING TERM 2009 COURSE AMATH 250 COURSE TITLE INTRODUCTION TO DIFFEREN
TIAL EQUATIONS SECTIONﬂS) 001 HELD WITH COURSE(S) ' N/A SECTIONS) OF HELD WITH COURSE(S) N/A DATE OF EXAM Monday, June 8th, 2009 TIME PERIOD 4:30  5:30 pm DURATION OF EXAM 60111111111363 NUMBER OF EXAM PAGES 6 pages (including this cover sheet) INSTRUCTOR D. IIarmsworth EXAM TYPE Closed book ADDITIONAL IEIATERIALS ALLOWED Scientiﬁc Calculator fr—
Name (print) l O M 5 ID Nllﬂlber \— f _ "F In" Signature Instructions: I. Print your name and ID Number on this
page and sign it. MARKING SCHEME [0 . Answer the questions in the space pro—
vided. Continue on the back of the page
if necessary. Show all your work. 3. The lest page may be removed and used
for 1011in work. 4. Your grade will be inﬂuenced by how
Clearly you express your ideas and by how
well you organize your solutions. AMATH 250 ~ Term Test #1 SpringTerm 2009 I. [10] 1. Solve the initial problem + By = 9! — 3—3;, 3,:(0) = l. 9mm ﬂax 'u a “new eoLun’FM: 'w: CM Cmdmc’f "he Sold?“ on ‘jr’: M C631 ,
LAL‘QL jwtg LILS 'gxﬂ "33L "31 §/I x'
PH C0. ZCxe, + gAl+3B+3C1€ : [ix—.6
Le. 3A1 4r 4 C631 :2 x he,ng ///
30 = f 91 q 35) 131‘} (DNA ED—L ) O
Awgﬁﬁo /
FRL r150:  "3" —
e e 3' Ce *gx[hxegx Page 2 of 6 AMATH 250  Term Test #1 SpringTerm 2009
[15] 2. Find the. general solution to the equation (1’.
(at:2 + 1) g + my : I3, and sketch the family of solutions.
.x "(LB “\5 \‘wmc. I 5}" 5:3 gum an 1 _ 38
h 0A f 't' } :3 *I T L3 __ T Q
5‘1 1H and. I I l .. 3
Wkﬂchg x943; 3. 1, d3" [8; utla*\ 1“ A“: 21:11 " No *Utrﬁi nsjm?\u\e5.
o Sotuans genius; “A. ekle'ROIwX > 5o\u¥w~ as jﬂ‘jod’ dino‘ as jLD—aa. Page 3 of 6 AMATH 250 — Tenn Test #1 SpringTeIIn 2009 (313 P
3. The Logistic Model of population growth uses the equation 2 (LP (1  E), where PO?) is the population at time t, and K is the “carrying capacity” (the maximum
sustainable population) and a is a proportionality constant. [5] (a) Show that by introducing appropriate dimensionless variables a and 1" this equation
0hr
can be written in the form d—T = :rr(1 — arr). tllxe \fmlables MA the} Amend“; on;
at t,{
[t]: 'r . 0
Tim was mt ea (imam; m.
[at = T‘
[K]: he.
We mung um. K 05 c. (waat‘iﬁrlsl'in PortiaRan; out ELK a; s. oLaraeierrshl ‘iim.
So] if." T: £1 and T: 0th @ —..... TM a? one new
6 (it‘dtdtavkﬁ J (D cmA ‘itmz, batonm5 : mic“ (eri)
cit Le. (Lil:TCO—ii)
an: [7] (13) Find the general solution to the nondimensionalized equation.
It’s sauna j 43 t J on @
nitTr]
\IJQ Mal 'l‘kt Firle 'QmEtU— [lemmfosiiion: l : £1: + B. G .L .L : "at
> 5th l l‘ﬁldﬁ l 7 T: (ll—Tile}  .. 'C_ t
:7 jn\n\—inllTll : I+L @ 1r ’rLJTe  (.9
 1'
ﬂ 9" Eljmc T: Cie : gt
i“Ti l¥ctet ":9 T,
1 : CIB Page 4 of 6 UMKLR 9pc]: eat )
i'T\ ' _ AMATH 250  Term Test #1 SpringTerrn 2009 [8] 4. Consider a cylindrical pipe, of radius a, containing a ﬂuid with dynamic viscosity ,a. If the
pressure is constant throughout the pipe then the ﬂuid will not flow, so to generate motion
tip (is:
is constant, call it 0:. Let’s also assume that the ﬂow 15 laminar; i.e. there is motion in only one direction. The speed of this ﬂowth Should be a function of the distance from the centre
of the pipe, 7', and it should also depend upon a, n, and a. Apply Bl_zcki11gha.1n’s H Theorem
to investigate the nature of this relationship. we need a nonzero pressure gradient, . For the sake of simplicity let’s suppose that this Note: [n] = fiIL"T“', and I remind you that pressure is force per unit area; that should
enable you to ﬁnd the dimensions of the pressure gradient 0:. \Jge have: [pﬂ1 METl _ : {gufm/wm} M [2.1 .}
“"K .__.—.._...—.. :
L 1 Llemj’fkl Eco3 ‘' Llll @ {a}; L
ErdzL .53. ghost} he, L\€£Ar that like“? Due. .3 hthth iiiMansion here, 50 Lee St‘wltl
\m 0%,, Jr, Cumtud 5'3: 2; Malawian dimsiuhleﬁ varLinks. @ \m can has, so, 1:}: Hm
(Sims. wért interest223‘ in soldan lin Lt) ) :3
ital“) M _ I. O
l‘i’ ﬁllol
°’ signii i Other Coffﬁei Cowa\a.s‘\ms cue. possihle, loud its, is emitslain 3m Easiest i2. 'ini't'f'ct’, glmg onlj “Ck
omens more than date) Page 5 of 6 AMATH 250  Term Test #1 SpringTerln 2009
(This page is for rough work) Page 6 of 6 ...
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This note was uploaded on 11/28/2009 for the course AMATH 250 taught by Professor Ducharme during the Spring '09 term at Waterloo.
 Spring '09
 Ducharme

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