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The University of British Columbia
MATH 215/255, Section 104
Midterm Exam I – October 2014
Signature
Name
Student Number
This exam consists of 4 questions worth 10 marks each. No notes nor calculators.
Problem
1.
max score
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 2014
Math 215/255 Midterm 1, Section 104
Page 2 of 5
(10 points) 1. Consider a fish population in a lake. Suppose some toxic substance flows into the lake so that
the rate of growth of fish is slowing down. We model such a population dynamics by
dp
= (12 − 3t2 )p.
dt
a) Find the general solution to the equation.
b) What is the maximum fish population and when does it occur if p(0) = 500?
October 2014
Math 215/255 Midterm 1, Section 104
(10 points) 2. a) Sketch the slope field and some typical solutions for
b) Draw the phase diagram for the equation
equilibrium points.
dx
dt
dx
dt
Page 3 of 5
= −x2 + x + 2.
= −x2 + x + 2 and indicate the stability of
2
c) For dx
dt = −x + x + 2 with initial condition x(0) = 0, use Euler’s method with step size
h = 0.5 to approximate x(1).
d) Find all possible numbers a so that the equation
dx
= ax2 + x + 2
dt
has two equilibrium points and the larger equilibrium is stable. Hint: Sketch parabola for
different values of a.
October 2014
Math 215/255 Midterm 1, Section 104
(10 points) 3. Show that r(x, y) = (xy)−1 is an integrating factor for
2
2
(3xy 3 ex +x)y 0 + 2x2 y 4 ex +y = 0,
and solve the equation for y(1) = 1.
Page 4 of 5
October 2014
Math 215/255 Midterm 1, Section 104
(10 points) 4. a) Find a general solution to
y 00 + 2y 0 − 3y = 0.
b) Solve
9y 00 − 12y 0 + 4y = 0,
y(0) = 1,
y 0 (0) = 0.
Page 5 of 5