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The University of British Columbia
MATH 215/255
Midterm Exam I – October 2014
Name
Student Number
Signature
Section Number
This exam consists of 4 questions worth 40 marks. No notes nor calculators.
Problem max score
1.
8
2.
12
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 .

October 2014
(8 points) 1.
(a) Solve y
0
= xy , y (1) = 1.
Math 215/255 Midterm 1 Page 2 of 5
(b) Use Euler’s method to approximate y (3) with step size h = 1.

October 2014 Math 215/255 Midterm 1 Page 3 of 5
(12 points) 2.
Newton’s law of cooling states that the rate of change in temperature of a body is equal to the temperature difference between the body and its surrounding environment, multiplied by a heat loss parameter. We place a body of initial temperature T
0 in an environment of temperature 10
◦
C, with heat loss parameter h = − 2.
(a) Find a general expression for the temperature of the body at any given time, in the case where T
0
= 50
◦
C. Describe the long-term behavior of the solution.
(b) Find a general expression for the temperature of the body at any given time in the case where T
0
= 5
◦
C. Describe the long-term behavior of the solution.
(c) Find the critical point of the autonomous differential equation you just solved, and based on questions (a) and (b) argue whether it is stable or unstable.

October 2014 Math 215/255 Midterm 1
(10 points) 3.
Find an integrating factor µ ( x ) so that the following equation is exact:
µ ( x ) y + µ ( x )( x
2 y − x ) dy dx
= 0 , y | x =1
= 3 .
Solve y as an implicit function of x .
Page 4 of 5

October 2014 Math 215/255 Midterm 1 Page 5 of 5
(10 points) 4.
Let α be a constant. Solve y
00 − 2 y
0
+ α (2 − α ) y = 0 for y (0) = 0 and y
0
(0) = 2. (Hint: You need to consider two cases which depend on α .)