Be sure this exam has 6 pages including the cover
The University of British Columbia
MATH 215/255
Midterm Exam II – November 2014
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Signature
Student Number
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This exam consists of 4 questions worth 10 marks each. No notes nor calculators.
Question
Points
1
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 2014
Math 215/255 Midterm Exam II
(5 points) 1. (a) Find a particular solution to y 00 + 3y = 9x2 + 3x.
(5 points)
(b) Find the inverse Laplace transform f (t) of F (s) =
4
.
s(s2 + 4)
Page 2 of 6
November 2014
Math 215/255 Midterm Exam II
Page 3 of 6
(5 points) 2. (a) Find a particular solution to y 00 + 2y 0 + y = 2e−x using the method of undetermined
coefficients.
(5 points)
(b) Find a particular solution to the same equation y 00 + 2y 0 + y = 2e−x , this time using the
method of variation of parameters.
November 2014
Math 215/255 Midterm Exam II
Page 4 of 6
3. Suppose the displacement x(t) of a damped mass-spring system subject to sinusoidal forcing
of amplitude F0 is modelled by:
x00 + 4x0 + 13x = F0 sin(t),
x(0) = 0,
x0 (0) = 1.
(5 points)
(a) Find the solution x(t) when F0 = 0 (no forcing).
(5 points)
(b) Now take F0 = 40, and find the steady periodic solution (the part of the solution x(t)
which remains as t → ∞). Do not find the transient part.
November 2014
Math 215/255 Midterm Exam II
Page 5 of 6
(5 points) 4. (a) Find the Laplace transform of the solution to the following initial value problem:
x00 − x0 + x = f (t),
where
f (t) =
(5 points)
x(0) = 0,
and x0 (0) = 1,
1, if 0 ≤ t < 1,
t, if t ≥ 1.
(b) Solve the initial value problem:
x00 + x0 = δ(t − 1),
x(0) = x0 (0) = 0.
November 2014
Math 215/255 Midterm Exam II
Page 6 of 6
Table of Laplace transforms
f (t) = L−1 {F (s)}
F (s) = L{f (t)}
1. 1
2. e−at
3. tn , n positive integer
4. sin(at)
5. cos(at)
6. sinh(at)
7. cosh(at)
8. u(t − a)
1
, s>0
s
1
, s > −a
s+a
n!
, s>0
n+1
s
a
, s>0
2
s + a2
s
, s>0
2
s + a2
a
, s > |a|
s2 − a2
s
, s > |a|
2
s − a2
e−as
, s>0
s
9. u(t − a)f (t − a)
e−as F (s)
10. e−at f (t)
Z t
11.
f (t − τ )g(τ )dτ
F (s + a)
F (s)G(s)
0
Z
t
0
F (s)
s
13. δ(t − a)
e−as
14. f (n) (t)
sn F (s) − sn−1 f (0) − ... − f (n−1) (0)
12.
f (τ )dτ
Variation of parameters
If y1 (x) and y2 (x) are two solutions of Ly = 0, then the particular solution of Ly = f (x) is
yp (x) = u1 (x)y1 (x) + u2 (x)y2 (x),
y1 u01 + y2 u02 = 0,
y10 u01 + y20 u02 = f (x).