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The University of British Columbia
MATH 215
Midterm Exam I – February 2010
Name
Student Number
Signature
This exam consists of 4 questions worth 10 marks each. No notes nor calculators.
Problem max score
1.
10
2.
10
3.
4.
10
10 total 40 score
1. Each candidate should be prepared to produce his library/AMS card upon request .
2. Read and observe the following rules :
No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action.
(a) Making use of any books, papers or memoranda, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
3. Smoking is not permitted during examinations .

February 2010 Math 215 Midterm 1 Page 2 of 5
(10 points) 1.
(a) Using the local existence theorem for first order equations (Theorem 2.4.2), decide if the equation d dt y = t
1 / 2
+ ( y − 1)
2
, y (0) = 1 , has exactly one solution near ( t
0
, y
0
) = (0 , 1). Explain your answer.
(b) Repeat part (a) for d dt y = t
2
+ ( y − 1)
1 / 2
, y (0) = 1 .
(c) Decide the interval of existence for the equation
(9 − t
2
) y
00
+ (sin t ) y
0
+ (ln t ) y = e t
, y (1) = 1 , y
0
(1) = 2 .

February 2010 Math 215 Midterm 1 Page 3 of 5
(10 points) 2.
The population of mosquitoes in a certain area increases at a rate proportional to the current population and, in the absence of other factors, the population becomes e times the original population every 10 days. There are 200,000 mosquitoes in the area initially, and predators
(birds, bats, and so forth) eat 20,000 mosquitoes per day. Determine the population of mosquitoes in the area at any day.

(10 points)
February 2010
3.
Solve
Math 215 Midterm 1 dy dx
3 x
2 y + y
2
= −
2 x 3 + 3 xy
, y (1) = − 2.
Page 4 of 5

February 2010 Math 215 Midterm 1
(10 points) 4.
Solve y
00
+ 2 y
0
+ 5 y = 0 , y (0) = 3 , y
0
(0) = 1.
Page 5 of 5