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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"wwwmmwamsmmwW4fist?I “fun‘51:;,.jT;Li"=*.“‘4‘ fulﬁl... a“; h" 3‘4"" 11.115: l A ‘ \u .'<.r:FMJ:‘ 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 phaseamplitude 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.
$5 ; ;_:>1tuﬁy>3€iﬂl’hrz why; m1 _ Continued on page 6 April 2001 MATH 256 Name —___— Page 6 of 12 pages 4. Consider the forced undamped mass/spring system, described by: (12.7: 2 . n
7122— + 0.10 a: : F0 [51n('yt)] (*) where n is an integer and we > O, 7 > 0. The following trigonometric identities will be useful: 2[sinA]2 = 1 — cos (2A)
2sinAcosB = sin(A +B) +sin(A— B). '_ (a) If n = 1, for what value of 'y will resonance occur? Explain. (b) If n = 2, for what value of ’y will resonance occur? Continued on page 7 April 2001 MATH 256 Name ___.__—_.__._.__ Page 7 of 12 pages (c) Find the two values of *y for which resonance will occur if n = 3. Continued on page 8 April 2001 MATH 256 Name W Page 8 Of 12 pages [15] 5
(a) Find the general solution of e\
3
u 93(75) + 8W),
* W)  31(75) 9;"
x
A
PF
v
 (b) Classify the origin, (0, O), as one of: spiral point, centre, saddle point or node. (c) ’Find the solution that satisﬁes the initial condition (a:(0),y(0)) = (1, ﬂ), where B is a
constant. For What value of ﬂ does ($(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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 Spring '08
 KEQINLIU
 Math, Differential Equations, Equations

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