math256_april2001

math256_april2001 - mu. »~—_. 44...: n - Be sure that...

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Unformatted text preview: mu. »~—_. 44...: n - Be sure that this examination has 12 pages including this cover The University of British Columbia Sessional Examinations - April 2001 Mathematics 256 Difierential Equations Closed book examination _ Time. 2% hours Name Signature Student Number____.__._.___ Instructor’sNarne Section Number Special Instructions: One 8%” X 11” two sided cheat sheet is permitted. No graphical or programmable calculators allowad. Basic scientific calculators allowed. Show your work in the spaces provided. Rules Governing Formal Examinations 1. Each candidate must be prepared to produce, upon request, a library/AMS card for identification. 2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. I 3. No candidate shall be permitted to enter the examination room after the expiration of one half hour from the scheduled starting time, or to leave during the first half hour of the examination. 4. Candidates suspected of any of the following, orsimilar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (30 Having at the place of writing any books, papers or memoranda, calculators, computers, audio or video cassette players or other, memory aid devices, other than those authorized by the examiners. 0") Speaking or communicating with other candidates. (‘3) Purposely exposing written papers to the View of other candidates. The plea of HCCident or forgetfulness shall not be received. 5- Candidates must not destroy or mutilate any examination material; must hand m all examination papers; and must not take any examination material from the . examination room without permission of theninvigilator. New"wwwmmwamsmmw-W4fist?I “fun-‘51:;,.j-T;Li"=*.“‘4‘ fulfil... a“; h" 3‘4"" 11.115: l A ‘ \u .'<.|r:FM-J:‘ April 2001 Marks [12] 1. MATH 256 Name Consider the initial value problem 1/05) =1+(y - t)2, 24%) = yo. (a) If u(t) = y(t) — t, find the initial value problem satisfied by u. 1 (b) Solve for y(t) when yo = You will want to use (a). 1 (c) Solve for y(t) when yo = 0. (Note: 6 is undefined.) Page 2 of 12 pages Continued on page 3 April 2001 MATH 256 Nanm —.____._______— Page 3 of 12 pages [13] 2. I Solve I . . ' ‘ y”(t) + 2y (t) + 5y(t) = 0 11(0) =1, y’(0) = 1 - fi- Give the solution in terms of a cosine with phase shift (i.e. in phase-amplitude form) and sketch it. Continued on page 4 April 2001 MATH 256 Name __.___._—___ Page 4 of 12 pages 4 . . . [15] 3. A spherical raindrop has mass m(t) = §7rr3(t) p, Where r(t) is the radius and p is the den51ty. It loses mass through evaporation, at a rateproportional to its surface area 305) = _47r'r2 d . . (a) Using the equation —tm(t) = —ks(t), and the condition r(0) = To, derive an expression d for the radius r(t) of the drop after time t. (b) Suppose that the raindrop falls from a cloud. Neglecting air resistance and using Newton’s . d third law, in the form Et—[m(t)u(t)] = m(t)g, where g = 9.8ms‘2, derive the following equation for the speed 1105) of the raindrop (take 1/ > 0 for downward motion): all/(t) 3(k/p) dt + [(k/pfi —r0] Continued on page 5 April 2001 MATH 256 Name W Page 5 of 12 pages I (c) Solve for 1/(t) assuming the raindrop is initially at rest. ((1) Ifirro = 2 x 10"3 m, and after 10 seconds r(t) = 1.4 x 10‘3 m, at what time will the raindrop completely evaporate? :3. $5 ; ;_-:>1t-ufiy>3€ifll’hrz why; m1 _ Continued on page 6 April 2001 MATH 256 Name —___— Page 6 of 12 pages 4. Consider the forced undamped mass/spring system, described by: (12.7: 2 . n 7122— + 0.10 a: : F0 [51n('yt)] (*) where n is an integer and we > O, 7 > 0. The following trigonometric identities will be useful: 2[sinA]2 = 1 — cos (2A) 2sinAcosB = sin(A +B) +sin(A— B). '_ (a) If n = 1, for what value of 'y will resonance occur? Explain. (b) If n = 2, for what value of ’y will resonance occur? Continued on page 7 April 2001 MATH 256 Name ___.__—_.__._.__- Page 7 of 12 pages (c) Find the two values of *y for which resonance will occur if n = 3. Continued on page 8 April 2001 MATH 256 Name W Page 8 Of 12 pages [15] 5- (a) Find the general solution of e\ 3 u 93(75) + 8W), * W) - 31(75)- 9;" x A PF v | (b) Classify the origin, (0, O), as one of: spiral point, centre, saddle point or node. (c) ’Find the solution that satisfies the initial condition (a:(0),y(0)) = (1, fl), where B is a constant. For What value of fl does ($(t), y(t)) converge to (0, 0) as t —> 00? Continued on page 9 15} April 2001 6. Page 9 of 12 pages MATH 256 Name A thin wire of length 1 has insulated lateral surfaces. Heat is generated in the wire at a constant rate by a current flowing through it. The equation describing changes in temperature in the wire is ut = um + 2. The ends of the wire are maintained at temperature 0 and initially the Whole wire is at this temperature. a) Find the steady state temperature distribution, v(9;), in the wire. b) Find the partial differential equation, bounday conditions and initial condition satisfied by the transient distribution, w(t,x) _=_= u(t, 113) - c) Find it by solving for w. Continued on page 10 April 2001 MATH 256 Name ___.________________.__ Page 11 0112 pages [15] 7. A rectangular drum is struck at time zero. The displacement from rest of the membrane satisfies utt=a2[um+uyy] forallt>0,0<a:<1,0<y<2, u(t,0,y) =u(t,l,y) =u(t,$,0) :u(t,x,2) = 0 for all t > 0,0 < z <17 0 < y < 2, u(0,x,y) = 0, mm, any): sin (ww)si'n(7ry) for all 0 < a: <17 0 < y < 2. Use separation of variables to find u(t, as, 3/). Continued on page 12 ...
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