TermTest1S09Solutions

TermTest1S09Solutions - UNIVERSITY OF WATERLOO TERM TEST #1...

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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 3-5) 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 *Utrfi-i nsjm?\u\e5. o Sotuans genius; “A. ekle'ROI-w-X > 5o\u¥w~ as jfl‘jod’ dino‘ as jL-D—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 Portia-Ran; out ELK a; s. oLaraeierrshl ‘i-im. So] if." T: £1 and T: 0th @ -—..... TM a? one- new 6 (it‘dtdtavkfi J (D cmA ‘itmz, baton-m5 : 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 @ nit-Tr] \IJQ- Mal 'l‘kt Firle 'QmEtU— [lemmfosii-ion: l : £1: + B. G .L .L : "at > 5th l l‘fildfi l 7 T: (ll—Tile} - .. 'C_ t :7 jn\n\—inll-Tll : 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] = fi-IL"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 MET-l _ : {gufm/wm} M [2.1 .} “"K .__.—.._...—.. : L 1 Llemj’fkl Eco-3 ‘-'- L-l-ll @ {a}; L ErdzL .53. ghost} he, L\€£Ar that like“? Due. .3 hthth iii-Mansion here, 50 Lee St‘w-ltl \m 0%,, Jr, Cumtud 5'3: 2; Malawian dimsiuhlefi var-Links. @ \m can has, so, 1:}: Hm (Sims. wért interest-223‘ in soldan lin- Lt) ) :3 ital“) M _ I. O l‘i’ fi-llol °’ sign-ii i Other Cofffiei Cowa\a.s‘\ms cue. possihle, loud its, is emits-lain 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.

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TermTest1S09Solutions - UNIVERSITY OF WATERLOO TERM TEST #1...

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