Be sure this exam has 11 pages including the cover
The University of British Columbia
Final Exam – December 2007
Mathematics 215/255, Ordinary Differential Equations
Name
Student Number
Circle Section: 101 Phan
Signature
102 Burghelea
103 Tsai
104 Flowers
105 Brydges
This exam consists of 8 questions worth 100 marks in total. No aids are permitted.
Problem
1.
max score
10
2.
10
3.
10
4.
10
5.
15
6.
15
7.
15
8.
15
total
100
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.
December 2007
Math 215/255 Final Exam, All Sections
Page 2 of 11
(10 points) 1. A bath contains water. Let the depth of water at time t be y = y(t) ins. At time t = 0 the
plug is pulled and water exits via the drain at the rate of 2y inches per minute.
(a) The bath is filled to an initial depth of 10 inches. Find the depth y as a function of time.
(b) Same problem as in the previous part except that at time t = 0 the tap is turned on and
water enters the bath at rate 3 inches per minute.
December 2007
Math 215/255 Final Exam, All Sections
Page 3 of 11
(10 points) 2. Consider
t + ye2ty + ate2ty
dy
= 0.
dt
(a) Determine a such that (1) is exact.
(b) Solve (1) when a has the value found in the previous part.
(1)
December 2007
Math 215/255 Final Exam, All Sections
Page 4 of 11
(10 points) 3. For the equation y 0 = 1 − x − y with initial condition y(0) = 2 write down the Euler recursion
with h = 0.1 for finding the solution approximately and use it to approximate y(0.2).
December 2007
Math 215/255 Final Exam, All Sections
(10 points) 4. (a) Find the general solution to y 00 − 2y 0 + 2y = 0.
(b) Find a particular solution to y 00 − 2y 0 + 2y = et cos t
(c) Find the general solution to y 00 − 2y 0 + 2y = et cos t.
Page 5 of 11
December 2007
(15 points) 5. (a) Solve
y 00
Math 215/255 Final Exam, All Sections
+
y0
(
t
=
0
(b) Find F (s) = L
n
1
√
t
if t ≥ 1
when y(0) = y 0 (0) = 0.
otherwise
o
√
√
√
, given that L{ t} = 12 πs−3/2 and F (1) = π.
(c) Find the inverse Laplace transform of
e−2s
.
(s + 1)2
Page 6 of 11
December 2007
Math 215/255 Final Exam, All Sections
Page 7 of 11
(15 points) 6. Consider
d
dt
x
y
=A
x(0)
y(0)
(a) Find the solution of (2) such that
(b) Find the solution of (2) such that
− 21
,
A=
−1
x(0)
1
=
.
y(0)
0
x
y
=
0
.
1
1
−
1
2
.
(2)
December 2007
Math 215/255 Final Exam, All Sections
(c) Find the matrix etA .
(d) Draw the phase portrait for (2) including directions of flow.
Page 8 of 11
December 2007
(15 points) 7. Consider
d
dt
Math 215/255 Final Exam, All Sections
x
y
=
2y + y 3
2x + 1
(a) Find the orbits.
(b) Find and classify equilibrium point(s).
(c) Draw the local phase plane portrait near the equilibrium point(s).
Page 9 of 11
December 2007
Math 215/255 Final Exam, All Sections
Page 10 of 11
(15 points) 8. For the following questions, fill in the answers in the boxes. No work need to be shown and no
partial credit will be given. Anything written outside of the boxes is ignored.
(a) Convert the equation
dy
d2 y
+ p + qy = 0,
2
dt
dt
y(0) = 1,
y 0 (0) = 2.
into an equivalent first order system of the form
d x1
?
x1 (0)
?
=
.
=
,
?
x2 (0)
?
dt x2
Answer =
(b) In the method of Picard iteration the equation
solved by a recursion of the form
Z
yn+1 (t) =? +
dy
dt
= y 2 , with initial condition y(0) = 7 is
?
? ds,
y0 (t) =?
?
Find y1 (t).
Answer =
(c) Is the solution to
dy
dt
= y 2/3 with y(0) = 1 unique?
Answer =
(d) Is the solution to
dy
dt
= y 2/3 with y(0) = 0 unique?
Answer =
(e) The Wronskian of two solutions of y 00 + 3y 0 + 2y = 0 is proportional to eat . Find a.
Answer =
December 2007
Math 215/255 Final Exam, All Sections
Table of Laplace transforms
f (t)
C
eat
tn , n positive integer
tp , p > −1
sin(at)
cos(at)
sinh(at)
cosh(at)
δ(t − c)
F (s) = L[f (t)]
C
s
1
s−a , s > a
n!
,s > 0
sn+1
Γ(p+1)
,
sp+1
a
,
s2 +a2
s
,
s2 +a2
a
,
s2 −a2
s
,
s2 −a2
e−cs
s>0
s>0
s>0
s > |a|
s > |a|
eat · sin(bt)
b
,
(s−a)2 +b2
s>a
eat · cos(bt)
s−a
,
(s−a)2 +b2
s>a
tn · eat , n positive integer
n!
,s
(s−a)n+1
>a
Hc (t)f (t − c)
e−cs
s , s>
e−cs F (s)
ect · f (t)
F (s − c)
Hc (t)
Rt
0
f (t − τ )g(τ )dτ
f (n) (t)
(−t)n f (t)
0
F (s)G(s)
sn F (s) − sn−1 f (0) − ... − f (n−1) (0)
F (n) (s)
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