THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
Term 1, 2020
MATH2018 / MATH2019
ENGINEERING MATHEMATICS 2D/E
(1) TIME ALLOWED – 3 hours
(2) TOTAL NUMBER OF QUESTIONS – 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) START EACH QUESTION ON A NEW PAGE.
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
YOU ARE TO COMPLETE THE TEST UNDER STANDARD EXAM
CONDITIONS, WITH HANDWRITTEN SOLUTIONS.
YOU WILL THEN SUBMIT ONE OR MORE FILES CONTAINING YOUR
SOLUTIONS. MAKE SURE YOU SUBMIT ALL YOUR ANSWERS.
ONE OF THE SUBMITTED FILES MUST INCLUDE A PHOTOGRAPH OF
YOUR STUDENT ID CARD WITH THE SIGNED, HANDWRITTEN
STATEMENT:
“I declare that this submission is entirely my own original work.”
YOU CAN DELETE AND/OR RELOAD FILES UNTIL THE DEADLINE.
Term 1, 2020 MATH2018 / MATH2019 Page 2
TABLE OF LAPLACE TRANSFORMS AND THEOREMS
The function g(t) is defined for all t ≥ 0, and its Laplace transform
G(s) = L{g(t)} =
∫ ∞
0
e−stg(t) dt
exists. The Heaviside step function u is defined by
u(t− a) =

0, for t < a,
1
2
, for t = a,
1, for t > a.
g(t) G(s) = L{g(t)}
1
1
s
t
1
s2
tν , ν > −1 ν!
sν+1
e−αt
1
s+ α
sin ωt
ω
s2 + ω2
cos ωt
s
s2 + ω2
u(t− a) e
−as
s
f ′(t) sF (s)− f(0)
f ′′(t) s2F (s)− sf(0)− f ′(0)
e−αtf(t) F (s+ α)
f(t− a)u(t− a) e−asF (s)
tf(t) −F ′(s)
Please see over . . .
Term 1, 2020 MATH2018 / MATH2019 Page 3
FOURIER SERIES
If f has period T = 2L, then
f(x) = a0 +
∞∑
n=1
(
an cos
npi
L
x+ bn sin
npi
L
x
)
,
where
a0 =
1
2L
∫ L
−L
f(x)dx
an =
1
L
∫ L
−L
f(x) cos
(
npi
L
x
)
dx, n ≥ 1,
bn =
1
L
∫ L
−L
f(x) sin
(
npi
L
x
)
dx, n ≥ 1.
LEIBNIZ RULE FOR DIFFERENTIATING INTEGRALS
d
dx
∫ v
u
f(x, t) dt =
∫ v
u
∂f
∂x
dt+ f(x, v)
dv
dx
− f(x, u) du
dx
.
MULTIVARIABLE TAYLOR SERIES
f(x, y) = f(a, b) + fx(a, b) (x− a) + fy(a, b)(y − b)
+
1
2
(
fxx(a, b)(x− a)2 + 2fxy(a, b)(x− a)(y − b)
+ fyy(a, b)(y − b)2
)
+ · · · .
VARIATION OF PARAMETERS
Consider a second-order linear differential equation
y′′ + p(x)y′ + q(x)y = f(x).
If y1(x) and y2(x) are linearly independent solutions of the the corresponding
homogeneous equation, then a solution of the inhomogeneous equation is given
by
y(x) = u1(x)y1(x) + u2(x)y2(x)
where
u1(x) = −
∫
y2(x)f(x)
W (x)
dx, u2(x) =
∫
y1(x)f(x)
W (x)
dx and W (x) = det
[
y1 y2
y′1 y
′
2
]
.
Please see over . . .
Term 1, 2020 MATH2018 / MATH2019 Page 4
SOME BASIC INTEGRALS∫
xn dx =
xn+1
n+ 1
+ C (n 6= −1)∫
dx
x
= ln |x|+ C∫
ekx dx =
ekx
k
+ C (k 6= 0)∫
ax dx =
ax
ln a
+ C (a 6= 1)∫
sin kx dx = −cos kx
k
+ C (k 6= 0)∫
sinh kx dx =
cosh kx
k
+ C (k 6= 0)∫
cos kx dx =
sin kx
k
+ C (k 6= 0)∫
cosh kx dx =
sinh kx
k
+ C (k 6= 0)∫
sec2 kx dx =
tan kx
k
+ C (k 6= 0)∫
cosec2 kx dx =
− cot kx
k
+ C (k 6= 0)∫
tan kx dx =
ln | sec kx|
k
+ C (k 6= 0)∫
sec kx dx =
1
k
(
ln | sec kx+ tan kx|)+ C (k 6= 0)∫
dx
a2 + x2
=
1
a
tan−1
(
x
a
)
+ C (a 6= 0)∫
dx√
a2 − x2 = sin
−1
(
x
a
)
+ C (a 6= 0)∫
dx√
x2 + a2
= sinh−1
(
x
a
)
+ C (a 6= 0)∫
dx√
x2 − a2 = cosh
−1
(
x
a
)
+ C (a 6= 0)∫ pi/2
0
sinn x dx =
n− 1
n
∫ pi/2
0
sinn−2 x dx (n = 1, 2, 3 . . . )∫ pi/2
0
cosn x dx =
n− 1
n
∫ pi/2
0
cosn−2 x dx (n = 1, 2, 3 . . . )
Please see over . . .
Term 1, 2020 MATH2018 / MATH2019 Page 5
Start a new page clearly marked Question 1
1. i) Find and classify the critical points of
f(x, y) = 2x3 − 15x2 + 36x+ 3y2 − 24y − 2.
Also give the function values at the critical points.
ii) Use the Leibniz rule to find
d
dx
∫ tan(x)
x
e−t
2
dt.
iii) Consider
y′′ − 2x−1y′ + 2x−2y = ln(x).
a) Show that the linearly independent functions y1 = x and y2 = x
2 are
solutions of the corresponding homogeneous equation.
b) Find a particular solution of the inhomogeneous equation.
iv) Suppose w = f(x, y) with x = u cosh(v) and y = u sinh(v). Show that(
∂w
∂u
)2
− 1
u2
(
∂w
∂v
)2
=
(
∂w
∂x
)2
−
(
∂w
∂y
)2
.
v) Consider f(x, y, z) = 2x2 − y2 + z2 at P0 = (1, 2, 3).
a) Calculate ∇f at P0.
b) Calculate the directional derivative of f at P0 parallel to the line
from (1, 2, 3) to (3, 5, 0).
Please see over . . .
Term 1, 2020 MATH2018 / MATH2019 Page 6
Start a new page clearly marked Question 2
2. i) Evaluate the double integral∫∫
Ω
(x2 + y2) dx dy,
where Ω is the triangle with vertices (0, 0), (1, 0), (1, 1).
ii) The solid S in R3 is bounded below by the paraboloid z = 6(x2 +y2) and
above by the paraboloid z = 7− (x2 + y2).
a) Show that the paraboloids meet on the circle x2 +y2 = 1 in the plane
z = 6.
b) Sketch the solid S in R3.
c) By evaluating an appropriate double integral in polar coordinates,
find the volume of S.
iii) A particle moves along a curve with parametric equations
x(t) = ln(t+ 1), y(t) = sin(t), z(t) = 3t,
where t is time.
Determine the speed of the particle at t = 0.
iv) Find the maximum and minimum values of f(x, y) = x2 +y2 on the circle
of radius 1 centred at (1, 0).
v) Consider the four points P (1, 2, 2), Q(1, 3, 3), R(2, 1,−1) and S(2, 2, 1)
in R3.
a) Find the vectors
−→
PQ,
−→
PR and
−→
PS.
b) Evaluate the scalar triple product
−→
PQ · (−→PR×−→PS).
c) What is the volume of the parallelepiped with one corner at P and
sides
−→
PQ,
−→
PR and
−→
PS ?
Please see over . . .
Term 1, 2020 MATH2018 / MATH2019 Page 7
Start a new page clearly marked Question 3
3. i) Consider the curve 3x2 + 8xy − 3y2 = 20.
a) Find an orthogonal matrix Q such that the transformation[
x
y
]
= Q
[
X
Y
]
reduces the quadratic form 3x2 + 8xy − 3y2 to principal axes.
b) Write the equation to the curve in terms of the new variables X and
Y .
c) Hence determine the points closest to the origin, both in the (X, Y ) co-
ordinates and in the original (x, y) coordinates.
ii) A body with position x = x(t) is subject to a restoring force −3x, a
frictional force −4x˙ and an applied force f(t). At t = 0, the body is at
rest in its equilibrium position. Thus,
x¨+ 4x˙+ 3x = f(t), x(0) = 0, x˙(0) = 0,
where x˙ = dx/dt and x¨ = d2/dt2. The applied force is as shown below.
t
f(t)
3
2
a) Find the Laplace transform F (s) = L{f(t)} of the applied force.
b) Find the function G(s) such that the Laplace transform of the solu-
tion, X(s) = L{x(t)}, has the form
X(s) = (1− e−3s)G(s).
c) Hence find g(t) such that
x(t) =
{
g(t), 0 < t < 3,
g(t)− g(t− 3), 3 < t <∞.
Please see over . . .
Term 1, 2020 MATH2018 / MATH2019 Page 8
Start a new page clearly marked Question 4
4. i) Define h : R→ R by
h(x) = x(pi − x) for 0 < x < pi,
together with
h(−x) = −h(x) and h(x+ 2pi) = h(x) for all x.
a) Sketch the graph of h(x) for −pi < x < 2pi.
b) Find the Fourier series expansion of h(x). Hint: use the properties
h(0) = 0 = h(pi) and h′′(x) = −2.
ii) The steady-state temperature distribution u = u(x, y) in a semi-infinite
slab of width pi/2 satisfies the Laplace equation,
∂2u
∂x2
+
∂2u
∂y2
= 0 for 0 < x <
pi
2
and 0 < y <∞.
The left side of the slab is held at 0 degrees, whereas the right side is
insulated. The base of the slab is held at a temperature h(x), and the
temperature tends to 0 at infinity. Thus,
u = 0 when x = 0 for 0 < y <∞,
∂u
∂x
= 0 when x =
pi
2
for 0 < y <∞,
u = h(x) when y = 0 for 0 < x <
pi
2
,
u→ 0 as y →∞, for 0 < x < pi
2
.
a) Show that if u(x, y) = F (x)G(y) is a solution of the Laplace equation,
then there is a separation constant k such that
F ′′ + kF = 0 and G′′ − kG = 0.
b) Use the boundary conditions at x = 0 and x = pi/2 to find all non-
trivial solutions F . You may assume that if k ≤ 0 then F ≡ 0, so
only the remaining case k = ω2 > 0 needs to be considered.
c) Solve for G to obtain a sequence of solutions un(x, y) of the Laplace
equation that satisfy each of the boundary conditions except u(x, 0) =
h(x).
d) Hence write down the solution if h(x) = 3 sinx− 2 sin 3x.
e) Also give the solution if h(x) = x(pi − x) as in part i) b) above.