THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
JUNE 2018
MATH2018 / MATH2019
ENGINEERING MATHEMATICS 2D /E
(1) TIME ALLOWED - 2 hours
(2) TOTAL NUMBER OF QUESTIONS - 4
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEPARATE BOOK
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ONLY CALCULATORS WITH AN AFFIXED "UNSW APPROVED" STICKER
MAY BE USED
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
JUNE 2018 MATH2018 / MATH2019
TABLE OF LAPLACE TRANSFORMS AND THEOREMS
g(t) is a function defined for all t 2'. 0, and whose Laplace transform
G(s) = L'.{g(t)} = 100 e-stg(t)dt
exists. The Heaviside step function u is defined to be
u(t- a)~ {:
g( t)
1
t
tm,m=O,l, ...
e-o:t
sin(wt)
cos(wt)
u(t - a)
f'(t)
f"(t)
e-o:t f (t)
J(t - a)u(t - a)
tf (t)
fort< a
fort> a
G(s) = L'.{g(t)}
1
s
1
s2
ml --
8m+l
1 - -
s+a
w
s2 +w 2
s
s2 +w 2
e-as
s
sF(s) - f(O)
s2F(s) - sf(O) - f'(O)
F(s + a)
e-as F(s)
-F'(s)
Page 2
JUNE 2018 MATH2018 / MATH2019 Page 3
FOURIER SERIES
If J(x) has period T = 2L, then the Fourier series of J(x) is defined by
where
ao + L ( an cos ( n; x) + bn sin ( 7 X))
n=l
ao = 2� j
L
J(x)dx,-L
an = ± 1-: f ( X) cos ( 7 x) dx, bn = 1 1-: f ( x) sin ( 7 X) dx
LEIBNIZ' RULE
d 1v(x) 1v(x) 8 f dv du
-d J(x, t)dt = -0 (x, t)dt + f ((x, v(x)) -(x) - f (x, u(x)) -d (x)X 11(x) 11(x) X dx X
MULTIVARIABLE TAYLOR SERIES
8f 8f J(x, y) = J(a, b) + (x - a) 8x (a, b) + (y - b) 8y (a, b)
+ - (x - a)2 -(a, b) + 2(x - a)(y - b)--(a, b) + (y - b)2-(a, b) + • • •1 ( 82 f 82 f 82 f ) 2! 8x2 8x8y 8y2
METHOD OF VARIATION OF PARAMETERS
Suppose that the homogeneous equation corresponding to the second order ODE
y" + p(x)y' + q(x)y = J(x)
has general solution Yh = Ay1 (x) + By2 (x). Then a particular solution to the above ODE is given by
y (x) = -y (x) J Y2(x)f(x) dx + y (x) J Y1(x)J(x) dxp l 1¥(x) 2 vV(x)
where W ( x), called the Wronskian of the functions y1 and y2, is defined by
HI ( X) = I Y1 ( X) Y2 ( X) I · y� (x) y;(x)
JUNE 2018 MATH2018 / MATH2019
SOME BASIC INTEGRALS
J xn+l xn dx = -- + C for n -=/= -1n+l
j idx=lnlxl+C
J ekx dx = ekx + Ck
J ax dx = -11 ax + C for a -=/= 1 na
J . cos(kx) sm(kx)dx=- k +C
J sin(kx) cos(kx)dx = --+Ck
J tan(kx) sec2(kx) dx = - - + Ck
j cosec2(kx) dx = -{ cot(kx) + C
J (k )d lnlsec(kx)I Ctan x x = k +
Page 4
j sec(kx) dx = {(ln I sec(kx) + tan(kx)I) + C
J 2 1 2dx = !tan-1 (:_) + Ca +x a a
J ✓ 1 dx = sin-1 (:_) + Ca2 - x2 a
J 1 dx = sinh-1 (:_) + C✓x2 + a2 a
J ✓ 1 dx = cosh-1 (:_) + C
x2 - a2 a
1� n -11� sinn x dx = -- sinn-2 x dx o n o
1� n -11� cosn x dx = -- cosn-2 x dx o n o
JUNE 2018 MATH2018 / MATH2019
Answer question 1 in a separate book
1. a) Consider the scalar field
¢(x, Y, z) = xez-l + cosy
and let F = '7¢.
i) Calculate F.ii) What is 'V x F?
Page 5
iii) Hence, or otherwise, calculate the line integral 1 F · dr along the
straight line path C from (1, 0, 1) to (5, 1r, 1).
b) Consider the function
J(x, y) = 2ey-l sinx.
i) Calculate the Taylor series expansion of f about the point ( i, 1) up
to and including linear terms.
ii) Determine the direction from the point (i, 1) for which the change
in f with distancea) is a minimum;/3) is zero.
c) Consider the polar curve r = l + sin 0 whose figure is given below.
Determine the area of the region enclosed by the curve by using a suitable double integral.
d) Consider the following ordinary differential equation
dy 2 sin(x2 )
dx + -;;Y = x2 ' x > 0 '
with initial condition y( �) = 0 . A student solves the ordinary differ­ential equation and writes the solution in an integral form, i.e.,
l jx y = 2 sin(t2 ) dt. X �
i) Verify that this function y satisfies the initial condition. ii) Use Leibniz' rule to verify that y satisfies the differential equation.

JUNE 2018 MATH2018 / MATH2019 Page 6
Answer question 2 in a separate book
2. a) An inhomogeneous Euler-Cauchy ordinary differential equation (ODE)is given by
2 d2y dy 2 x dx2 - 3x dx + 3y = 2x X > 0 .You are given that Yi = x and y2 = x3 are solutions to the corresponding homogeneous Euler-Cauchy ODE. You do not have to check this. i) Calculate the 1Nronskian of Yi and y2.ii) Use the method of Variation of Parameters to determine a particularsolution YP for the inhomogeneous Euler-Cauchy ODE.b) A real symmetric 3 x 3 matrix A has eigenvalues denoted by Ai , ,\2 , and ,\3 . ·we define a quadratic surface
X'Ax � 12 where x cc m
A student is given the following information about A: • trace(A) = 0,
• Ai = 2 and ,\3 = 4 with associated eigenvectors, respectively,
i) What is the value of the remaining eigenvalue, namely ,\2?ii) Write down the equation of the quadratic surface, relative to theprincipal axes of the surface.iii) ·write down a vector v2 that is orthogonal to both eigenvectors viand v3. iv) What is the relationship between ,\2 and v2 ? Give reasons for youranswer.v) Hence determine an orthogonal matrix P which diagonalises thematrix A such that p-iAP = D where Dis a 3 x 3 diagonal matrix.vi) Hence determine the matrix A.c) The function f = Jxl is defined on the interval [-1r, 1r). We extend f toa periodic function of period 21r.i) Sketch the function f on the interval -31r :S: x :S: 31r.ii) Determine the Fourier coefficient a0 for f.iii) Determine the Fourier coefficient an for f, n = l, 2, 3, .. ..iv) Determine the Fourier coefficient bn for f, n = l, 2, �' . . ..
JUNE 2018 MATH2018 / MATH2019
Answer question 3 in a separate book
3. a) Find
i) £ {te-t sin(3t)};
ii) ,e-1 { s + l }s2 + 4s + 5
b) The function f ( t) is defined for t 2 0 by
J(t) = {1, o,
0::; t < l,
t 2 1.
i) Express f ( t) in terms of the Heaviside function.
Page 7
ii) Hence or otherwise find£ {f(t)}, the Laplace transform of f(t).
c) Solve the differential equation
y"-4y'+4y=f(t), t>O,
subject to the initial conditions y(O) = 1 and y'(O) = 0, where J(t) is
given in part b).
JUNE 2018 MATH2018 / MATH2019 Page 8
Answer question 4 in a separate book
4. a) A student wants to use the method of Lagrange multipliers to find the
point on the surface
x2 - xy + y2 - z2 = 1
nearest to the origin. Write down the algebraic equations the student
needs to solve in order to find this point. You do not have to solve these
equations.
b) The steady state temperature u( x, y) in a slab of length a and width b
satisfies the equation
[J2u 82u
ax2 + By2
= 0 for 0 < x < a, 0 < y < b,
with boundary conditions
u(0, y) = v,(a, y) = 0,
u(x, 0) = 0, u(x, b) = 71,
y
bf---- ------
0 < y < b,
0 < X < a.
Q�---------...L..__ X
a
i) Assuming a solution of the form u(x, y) = X(x)Y(y), show that
for some constant k.
X" Y" -=--=k
X y
ii) Write down the ordinary differential equations and the associated
boundary conditions for X ( x) and Y (y).
iii) Consider all values of k, that is, k = p2 > 0, k = 0, k = -p2 < 0 and
solve for X(x).
iv) With the values of k obtained in iii) which yield non-trivial solution
X(x), solve for Y(y).
v) Write down the general solution of u(x, y).
END OF EXAMINATION 