Be sure this exam has 6 pages including the cover
The University of British Columbia
MATH 215
Midterm Exam II – March 2010
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.
March 2010
Math 215 Midterm Exam II
Page 2 of 6
(5 points) 1. (a) Use the method of reduction of order to find a second solution y2 (t) of the equation
t2 y 00 − 4ty 0 + 4y = 0,
t > 0;
which is independent of y1 .
(5 points)
(b) Find a particular solution of the equation
y 00 − 2y 0 = 4t.
y1 (t) = t
March 2010
Math 215 Midterm Exam II
Page 3 of 6
2. Suppose the motion of a certain mass-spring system satisfies the differential equation
u00 + 2u0 + 5u = 10 cos t,
u(0) = 3,
u0 (0) = −4,
with units m, kg, and s. Here u(t) is the displacement from the equilibrium position.
(7 points)
(a) Determine the position u(t) at any time t.
(3 points)
(b) Identify the transient and steady state parts of the solution. Find the amplitude of the
steady state part.
March 2010
Math 215 Midterm Exam II
(10 points) 3. Use Laplace transform to find the solution of the initial value problem
(
1 if 0 ≤ t < 3
y 00 + 2y 0 + y =
0 if t ≥ 3
y(0) = 0,
y 0 (0) = 0.
Page 4 of 6
March 2010
Math 215 Midterm Exam II
Page 5 of 6
(7 points) 4. (a) Use Laplace transform to find the solution of the initial value problem
y 00 + 4y 0 + 5y = δ(t − 3),
y(0) = 1, y 0 (0) = 0.
Z
(3 points)
(b) Find the Laplace transform of the function f (t) =
0
t
sin(t − τ )e−2τ dτ .
March 2010
Math 215 Midterm Exam II
Page 6 of 6
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)2 + b2
11. tn eat , n positive integer
n!
,
(s − a)n+1
12. uc (t)
e−cs
,
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
s>a
s>a
s>0
1 s
F
,
c
c
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)