Be sure this exam has 6 pages including the cover
The University of British Columbia
MATH 215/255, Section 102
Midterm Exam II – November 2009
Name
Signature
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.
November 2009
Math 215/255 Section 102 Midterm Exam II
(5 points) 1. (a) Find the general solution of the equation
y 00 + 2y 0 + y = 0.
(5 points)
(b) Find a particular solution of the equation
y 00 + 2y 0 + y = 2e2t .
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November 2009
Math 215/255 Section 102 Midterm Exam II
Page 3 of 6
(7 points) 2. (a) A spring is stretched 0.2 m by a force of 5 N. (1 N = 1 kg·m
.) A mass of 1 kg is hung
s2
from the spring and is also attached to a viscous damper that exerts a force of 18 N
when the velocity of the mass is 3 m/s. If the mass is set in motion from its equilibrium
position with a downward velocity of 0.4 m/s, determine its position u(t) at any time t.
Here u is the displacement from the equilibrium position measured positive downward.
(The mass spring system has equation mu00 + γu0 + ku = 0 where m is the mass attached
to the spring, γ is the damping coefficient so that the damping force is given by −γu0 ,
and k is the spring constant so that the force when the spring is stretched by u is −ku.)
(3 points)
(b) Find the quasi frequency µ and the ratio of µ to the natural frequency ω0 of the
corresponding undamped motion. (When γ 2 < 4km, the quasi frequency µ is given by
1 p
µ=
4km − γ 2 . The natural frequency is the same quantity when γ = 0.)
2m
November 2009
Math 215/255 Section 102 Midterm Exam II
(7 points) 3. (a) Use Laplace transform to find the solution of the initial value problem
(
4 if 0 ≤ t < π
y 00 + 4y =
0 if t ≥ π
y(0) = 1,
y 0 (0) = 6.
Z
(3 points)
(b) Find the Laplace transform of the function f (t) =
0
t
(t − τ )2 eτ dτ .
Page 4 of 6
November 2009
Math 215/255 Section 102 Midterm Exam II
(6 points) 4. (a) Use Laplace transform to find the solution of the initial value problem
y 00 + 2y 0 + y = δ(t − 3),
(4 points)
(b) Find the inverse Laplace transform of
y(0) = 1, y 0 (0) = 0.
F (s) = e−3s
s2
s
.
+ 4s + 5
Page 5 of 6
November 2009
Math 215/255 Section 102 Midterm Exam II
Table of Laplace transforms
f (t) = L−1 {F (s)}
1. 1
2. eat
3. tn , n positive integer
4. tp , p > −1
5. sin(at)
6. cos(at)
7. sinh(at)
8. cosh(at)
9. eat sin(bt)
F (s) = L{f (t)}
1
, s>0
s
1
, s>a
s−a
n!
, s>0
n+1
s
Γ(p + 1)
, s>0
sp+1
a
, s>0
2
s + a2
s
, s>0
s2 + a2
a
, s > |a|
2
s − a2
s
, s > |a|
2
s − a2
b
, s>a
(s − a)2 + b2
10. eat cos(bt)
s−a
, s>a
(s − a)2 + b2
11. tn eat , n positive integer
n!
,
(s − a)n+1
12. uc (t)
e−cs
, s>0
s
13. uc (t)f (t − c)
e−cs F (s)
14. ect f (t)
F (s − c)
15. f (ct)
Z t
16.
f (t − τ )g(τ )dτ
1 ³s´
F
,
c
c
s>a
c>0
F (s)G(s)
0
17. δ(t − c)
e−cs
18. f (n) (t)
sn F (s) − sn−1 f (0) − ... − f (n−1) (0)
19. (−t)n f (t)
F (n) (s)
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