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MATHEMATICS 2C03
.
DURATION:
60 MINUTES
McMASTER UNIVERSITY TERM TEST 2 v1
Dr. G. S. K. Wolkowicz
March 19, 2015
THIS TEST PAPER INCLUDES 10 PAGES, 6 QUESTIONS. IT IS POSSIBLE TO OBTAIN A
TOTAL OF 50 MARKS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF
THE PAPER IS COMPLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF THE
INVIGILATOR.
SPECIAL INSTRUCTIONS: Check to see that you have all the pages and that no pages are
blank. The last two pages can be used for rough work or for continuation of a problem if you run
out of room. No other paper is allowed.
You are NOT permitted to have any ELECTRONIC DEVICES of any kind, including cell
phones.
NO NOTES OR AIDS OR PIECES OF PAPER OF ANY KIND (other than that
distributed by the invigilator) ARE PERMITTED.
You must print your name and ID number at the top of each page in the space provided as well
as on this page below.
NAME:
ID #:
Questions
Multiple Choice
4
5
6
TOTAL
Mark
Out of
12
9
18
11
50
GOOD LUCK!
Continued on next page.
Page 1 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
MULTIPLE CHOICE
There are 3 multiple choice questions. All questions in this section have the same value. A
correct answer scores 4 and an incorrect answer scores zero. Record your answer by circling ONE
and ONLY ONE of the letters. Ambiguous answers will be considered as incorrect.
1. The general solution of
x2 y 00 (x) + 5xy 0 (x) + 4y(x) = 0, valid for x > 0,
is given by:
(A) y(x) = c1 x−2 + c2 x−4 , c1 , c2 arb.
(B) y(x) = c1 ln(x) + c2 (ln(x))2 , c1 , c2 arb.
(C) y(x) = c1 ln(x) + c2 x ln(x), c1 , c2 arb.
(D) y(x) = c1 x−2 + c2 x−2 ln(x), c1 , c2 arb.
(E) y(x) = c1 x−2 + c2 x−4 ln(x), c1 , c2 arb.
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Page 2 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
2. Which of the following is the inverse Laplace Transform of
7s − 5
.
s2 + 10s + 41
(A) e−5t (7 cos(4t) − 10 sin(4t))
(B) e−5t (7 cos(4t) − 9 sin(4t))
(C) e−5t (7 cos(4t) − 5 sin(4t))
(D) e−5t (7 cos(4t) + 5 sin(4t))
(E) e−5t (7 cos(4t) + 7 sin(4t))
Table of Laplace Transforms
L{f (t)} = F (s)
1
L{1} = 1s ; L{ekt } = s−k
;
β
s
n!
n
L{sin(βt)} = s2 +β 2 ; L{cos(βt)} = s2 +β 2 ; L{t } = sn+1
, n ≥ 1 integer;
kt
n
n (n)
L{e f (t)} = F (s − k); L{t f (t)} = (−1) F (s);
L{δ(t − a)} = e−as , a ≥ 0;
L{f (n) (t)} = sn F (s) − sn−1 f (0) − sn−2 f 0 (0) − ... − sf (n−2) (0) − f (n−1) (0);
(
0,
t < τ,
L{u(t − τ )f (t − τ )} = e−τ s F (s), τ ≥ 0, where u(t − τ ) =
1,
t ≥ τ,
is the unit step function or Heaviside function;
R
T
e−st f (t) dt
If f (t + T ) = f (t) for all t, L{f (t)} = 0 1−e−sT ;
Rt
L{(f ∗ g)(t)} = L{ o f (t − v)g(v) dv} = F (s)G(s);
R ∞ −u t−1
L{tr } = Γ(r+1)
,
r
∈
R
&
r
>
−1
where
Γ(t)
=
du, t > 0.
r+1
0 e u
s
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Page 3 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
dy(x)
= P (x)y 2 (x) + Q(x)y(x) + R(x) is called a
dx
generalized Riccati equation. If one solution u(x) is known, the substitution
3. An equation of the form
y(x) = u(x) +
1
v(x)
reduces this differential equation to which linear differential equation.
(A)
(B)
(C)
(D)
(E)
dv(x)
dx + (P (x) + Q(x))v(x) = −P (x)
dv(x)
2
dx + (u (x)P (x) + 2u(x)Q(x))v(x) = −P (x)
dv(x)
2
dx + (u (x)P (x) + 2u(x)Q(x) + R(x))v(x) = −P (x)
dv(x)
2
dx + (2u (x)P (x) + 2u(x)Q(x) + R(x))v(x) = −P (x)
dv(x)
dx + (2u(x)P (x) + Q(x))v(x) = −P (x)
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Page 4 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
COMPLETE ANSWER QUESTIONS
Questions 4-6 You must show your work to to receive full credit. You will be
graded on the clarity and presentation of the solution, not just upon whether or not
you obtain the correct solution. The value of each question is given in the margin.
4. Using the Method of Variation of Parameters, find a particular
solution of the differential equation
[9 marks]
x2 y 00 − 4xy 0 + 6y = x3 + 1,
x>0
given that y1 (x) = x2 and y2 (x) = x3 are two linearly independent solutions of
the associated homogeneous problem.
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Page 5 of 10
Mathematics 2C03 Test 2
[18 marks]
Name:
Student ID:
5. Consider the differential equation
d4 y
d3 y
d5 y
− 8 4 + 16 3 = 4x2 + sin(4x) + e4x .
5
dx
dx
dx
(i)
(ii)
[4 marks]
Find the general solution of the associated homogeneous equation.
If the Method of Undetermined Coefficients is to be used
to find a particular solution of the nonhomogeneous differential equation,
what is the correct form to use to find a particular solution? (DO NOT
SOLVE.)
[4 marks]
(iii) [10 marks] Find the general solution of the second-order linear differential
equation
y 00 + 2y 0 = 12x2 + 4.
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Page 6 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
Continuation of solution to problem 5.
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Page 7 of 10
Mathematics 2C03 Test 2
[11 marks]
Name:
Student ID:
6. Using the Method of Reduction of Order, find the general solution of
xy 00 + (1 − 2x)y 0 + (x − 1)y = ex ,
x > 0,
given that y1 (x) = ex is a solution of the associated homogeneous problem.
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Page 8 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
Continuation of solution to problem 6.
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Page 9 of 10
Mathematics 2C03 Test 2
Name:
Student ID:
Continuation of solutions.
If you need to continue the solution to a problem here, mark clearly on the page
where the problem is stated that the solution is continued here (i.e., page 10) and
on this page mark clearly which problem you are continuing.
Last Page
∗T HE EN D∗
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