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(math150)[2004](f)final~PPSpider ^_10462.pdf
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HKUST
MATH150 Introduction to Di.erential Equations
Final Examination (Version White) Name:
15th December 2004 Student I.D.:
8:30C10:30 Tutorial Section:

Directions:
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Writeyour name,IDnumber, and tutorial sectionin the space providedabove.

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DO NOT open the exam until instructed to do so.

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When instructedtoopenthe exam,check thatyouhave,in additionto thiscover page, 11 pages of questions.

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Turn o. all mobile phones and pagers during the examination.

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Thisisa closedbook examination.

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You are advised to try the problemsyou feel more comfortable with .rst.

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You maywrite onboth sides of the examination papers.

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There are8 multiplechoice questions. DO NOT guess wildly! Ifyou do not have con.dence inyour answer leave the question blank. Eachincorrectly answered question will result in a 0.5 point deduction.

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For the short and long questions, you must show the working steps of your answers in order to receive fullpoints.

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Cheating is a serious o.ense. Students caught cheating are subject to a zero score as well as additionalpenalties.


Question No. Points Out of
Q. 1-8 32
Q. 9-14 32
Q. 15 18
Q. 16 18
Total Points 100

Part I: Eachcorrect answerin the answer box for the following8 multiplechoice questionsis
worth4point. DO NOT guess wildly! Ifyoudo nothave con.denceinyour answer leave the
answerbox blank. Each incorrectly answered question will resultina 0.5point deduction.

Question 1 2 3 4 5 6 7 8 Total
Answer

1. Whichofthefollowing functionscanbeusedasanintegrating factortoturnthefollowing non-exact equation into an exact equation?
dy
(3ycos x . xy sin x)+2xcos x = 0?
dx
222 2
(a) x(b) xy (c) y(d) xy(e) xy
2. For a simple RLC series electrical circuit with R =1/5 ohm, L = 1 henry, and C farads, the di.erential equation for the current I through the circuit is
d2I CdI
C ++I =0 .
dt2 5 dt
Pickthe largest possibleC fromthefollowingwithwhichthe currentofthe circuitwillkeepchanging its direction (oscillates) as t .
(a) 260 (b) 100 (c) 80 (d) 62 (e) 15
3. On which of the given intervals will the following initial value problem have a unique, continuous, solution?
. dx1 2tt
.. e.. x1 .. tan t .. x1(3) .. 2 .
dt t.2
=+ , = ,dx2 tet tx2 t +t2 x2(3) .3
dt
3 3
(a) .2<t< 2 (b) <t< 3 (c) 2 <t< (d) <t< (e) 2 <t< 5
2 222
1
4. Determine the inverse Laplace transform of the function
(s+1)2 .
.t
(a) t (b) te.t (c) t2 (d) t2 e(e) tet
5. Consider the nonhomogeneous equation
.3t
y .. +6y . +9y = (2t +t4)e.
Bythe methodof undeterminedcoe.cients, thereisa solutiontothe equation whichisofthe form y = Q(t)e.3t where Q(t)isapolynomial. The degreeofQ(t)is:
(a)2 (b)3 (c)4 (d)5 (e)6
6. The following intial value problem of a .rst order linear system
. x=3x . 2y, x(0) = 1,
y= .3x +4y, y(0) = .2
canbe convertedinto an initialvalue problemofa 2nd order di.erential