Final Exam
MATH 150: Introduction to Ordinary Differential Equations
J. R. Chasnov
15 December 2005
Answer ALL questions
Full mark: 90; each question carries 10 marks.
Time allowed – 2 hours
Directions
– This is a closed book exam. You may write on the front and back of the exam papers.
Student Name:
Student Number:
Question No.
(mark)
Marks
1
(10)
2
(10)
3
(10)
4
(10)
5
(10)
6
(10)
7
(10)
8
(10)
9
(10)
Total
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Question 1
Score:
(a) (8 pts) Solve the initial value problem for
x
≥
0:
dy
dx
=
2 cos 2
x
3 + 2
y
,
y
(0) =
-
2
.
(b) (2 pts) Determine where the solution attains its minimum value. What is the value of y at its minimum?
– 1 –
Question 2
Score:
(a) (6 pts) With
λ
,
a
and
b
all positive, find the solution
x
=
x
(
t
) for
t
≥
0 of the following initial value
problem:
˙
x
+
λx
=
a
+
be
−
λt
;
x
(0) = 0
.
(b) (2 pts) Determine the value of
t
at which
x
(
t
) is maximum.
(c) (2 pts) Determine
x
as
t
→ ∞
.
– 2 –
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Question 3
Score:
Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the
difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of
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- Equations, Constant of integration, Boundary value problem
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