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(math150)[2007](s)final~PPSpider^_10465.pdf
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HKUST MATH150 Introduction to Di.erential Equations
Final Examination (Version A) Name: 26th May 2007 Student I.D.: 08:30C10:30 Tutorial Section:
Directions:
.
Write your name, ID number, and tutorial section in the space provided above.
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DO NOT open the exam booklet until instructed to do so.
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Wheninstructed to openthe exambooklet, check thatyouhave,in additiontothis coverpage, 12 pages of questions.
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Turn o. all mobile phones and pagers during the examination.
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This is a closed book examination.
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You are advised to try the problems you feel more comfortable with .rst.
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You may write on both sides of the examination papers.
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Thereare8 multiplechoice questions. DO NOT guess wildly! If you do not have con.dence in your answer leave the question blank. Each incorrectly 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 full points.
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You must not possess any written or printed papers that contains information related to this examination.
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Cheating is a serious o.ense. Students caught cheating are subject to a zero score as well as additional penalties.
Question No.
Points Out of
Q. 1-8
24
Q. 9-14
38
Q. 15
17
Q. 16
21
Total Points
100
Part I: Fill in answer in the allocated answer box for the following 8 multiple choice ques-tions is worth 3 points each. DO NOT guess wildly! If you do not have con.dence in your answer leave the box blank. Each incorrectly answered question will result in a
0.5 point deduction.
Question 12345678 Total Answer
1. Matchthegraph ofthedirection .eld(slope.eld) tothedi.erential equation.
2.5
dy
(a) = x .2y 2
dx 1.5
dy 1
(b) = x +2y
0.5
dx
0
dy .0.5 (c) =1.x .2y
dx .1
.1.5
dy
dx
(d) =2.y .2
.2.5
dy
(e) = x .2y+1
dx
2. Which of the following di.erential equations is/are linear equation(s)?
d3y
t
(I) + y = e
dt3
dy
(II) +ty2 =0
dt
d2y
(III) + sin(t + y)= sint
dt2
(a) (I) (b) (II) (c) (III) (d) (I)and(II) (e) (II)and(III)
3. Determine the interval in which the solution of
(t2 .9)y . +(lnt)y =2t, y(1)=2
is certain to exist.
(a) t< 3 (b) 3<t (c) .3 <t< 3 (d) 0 t 3 (e) 0<t< 3
y
4. Consider the di.erential equation
dy
M(x,y)+N(x,y)=0. (.)
dx
Which ofthefollowingis/are su.cient condition(s) for(.)to be exact.
.M .N
(I) =.
.x .y
.F .F
(II) There exists F(x,y)such that = M and = N.
.x .y
.M .N
(III) =0 and =0.
.y .x
(a)
(II)only
(b)
(II)and(III) only
(c)
(I)and(II) only
(d)
(I)and(III) only
(e)
(I),(II)and(III)
5. Consider the following di.erential equation 1dy
+ y =1
t dt
22
(a)
Integrating factor =e t 2 ;General solution = 1+Ce.t 2
.t;
(b)
Integrating factor =eGeneral solution = C
22
.
(c) Integrating factor =e t 2 ;General solution = 1+Ce t 2