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(MATH150)[2007](s)midterm~kkleeab^_85431.pdf
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HKUST
MATH150 Introduction to Di.erential Equations Mid-Term Examination (Version A) 23rd March 2007 Name: Student I.D.:
19:00C20:30 Tutorial Section:

Directions:
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DO NOT open the exam until instructed to do so.

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All mobile phones and pagers should be switched o. during the examination.

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Write your name, ID number, and Section in the space provided above.

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When instructed to open the exam, check that you have 9 pages of questions in addition to the cover page.

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This is a closed book examination.

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Graphical calculators are NOT allowed.

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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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There are 8 multiple choice questions. DO NOT guess wildly! Leave the question blank if you do not have con.dence in your answer. 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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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 32
Q. 9-15 50
Q. 16 18
Total Points 100

Part I: Each correct answer for the following 8 multiple choice questions is worth 4 point. 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.
Question 1 2 3 4 5 6 7 8 Total
Answer

1. Match the graph of the slope.eld to the di.erential equation. [2 pts]

dy dydy
(a) = x . y (b) = x + y (c) =1 . x . y
dx dxdx
dy dy

(d) =1 . y (e) = x . y +1
dx dx
2. A tank contains 240 litres of a salt solution, the actual salt content in the tank being 20 kgs. A new salt solution which contains 0.2 kgs of salt per litre is now pumped into the tank at the rate of 3 litres per minute. The solution is continually well-stirred and pumped out at the same rate of 3 litres per minute (so that the total amount of solution in the tank stays at 240 litres). Set up a di.erential equation for y(t) = the amount of salt (in kgs.) in the tank at time t (t being the number of minutes after one starts pumping in the new solution).
3t(a) y. =0.6t (b) y . =0.6 . 3y (c) y . =0.6t .
240 3y
(d) y. =0.2t . 3y t (e) y . =0.6 . 240
3. If y1 and y2 are linearly independent solution of ty.. . y. + (3t.1 + t)y = 0 and if W (y1,y2)(1) = 3, .nd the value of W (y1,y2)(2)
2
(a)1 (b)3e(c) 6 (d) 12 (e) 3ln2
4. Suppose f(t) and g(t) are linearly independent functions on R. Which of the following statement(s) must be true?
(I)
W (f, g) > 0

(II)
f(t)+ g(t) and f(t) are linearly independent on R.


(III) There does not exist any real number k such that kf(t)= g(t) for all t.
(a)
(I) only

(b)
(III) only

(c)
(I) and (II) only

(d