Be sure this exam has 5 pages including the cover
The University of British Columbia
MATH 215/255
Midterm Exam I – October 2015
Name
Signature
Student Number
Course Number
Circle Section:
101 Zhao
102 Tsai
103 Kolokolnikov
104 Zhao
This exam consists of 4 questions worth 40 marks. No notes nor calculators.
Problem
Points
1
10
2
10
3
10
4
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 2015
Math 215/255 Midterm 1
Page 2 of 5
1. Consider the following initial value problem:
( 0
y = xy − y,
y(2) = 2.
(6 points)
(a) Find the explicit solution to the problem (1).
(4 points)
(b) Use Euler’s method to approximate y(4) to the problem (1) with the step size 1.
(1)
October 2015
Math 215/255 Midterm 1
Page 3 of 5
2. A solution that contains 5 grams/liter of salt is being poured into a tank at a rate of 1
liters/minute. At the same time, the solution in the tank flows out at a rate of 2 liters per
minute. Initially, the tank contains 50 liters of pure water. Let y(t) be the amount in grams
of salt in the tank at time t minutes, then y(t) satisfies
y0 +
2
y = 5,
50 − t
y(0) = 0.
(2 points)
(a) How long will it take for the tank to be empty?
(8 points)
(b) How much salt will there be in the tank when 20 liters remain?
October 2015
Math 215/255 Midterm 1
Page 4 of 5
3. Let k be a real constant, consider the following differential equation:
(2x2 + ky) + (x2 y − x)y 0 = 0,
for x > 0.
(2)
1
is an integrating factor for the equation (2).
x2
(4 points)
(a) Find the value of k such that r(x) =
(6 points)
(b) When k takes the value found in the previous part, find the solution to the problem (2)
with the initial value condition y(1) = 2. (You can keep the implicit form of the solution)
October 2015
Math 215/255 Midterm 1
Page 5 of 5
(7 points) 4. (a) Consider the following autonomous differential equation:
y 0 = y(y − 1)2 (y + 2).
(3)
Draw the phase diagram of the problem (3), and classify the critical points of the
problem (3) stable or unstable.
(3 points)
2
(b) It’s easy to verify that y1 (x) = x and y2 (x) = x + 2e−x are two solutions to the
differential equation y 0 = 1 + 2x2 − 2xy. Let g(x) be the solution to the initial value
problem:
( 0
y = 1 + 2x2 − 2xy,
y(0) = 1.
Find the limit lim
x→∞
g(x)
and explain why. (Hint: Existence and uniqueness theorem)
x
(4)