THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS AND STATISTICS

JUNE 2008

MATH2019

ENGINEERING MATHEMATICS 2E

(1) TIME ALLOWED – 2 hours

(2) TOTAL NUMBER OF QUESTIONS – 4

(3) ANSWER ALL QUESTIONS

(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 THE PROVIDED ELECTRONIC CALCULATORS MAY BE USED

All answers must be written in ink. Except where they are expressly required pencilsmay only be used for drawing, sketching or graphical work.

JUNE 2008 MATH2019 Page 2

TABLE OF LAPLACE TRANSFORMS AND THEOREMS

g(t) is a function defined for all t ≥ 0, and whose Laplace transform

G(s) = L(g(t)) =

∫ ∞

0

e−stg(t)dt

exists. The Heaviside step function u is defined to be

u(t− a) =

0 for t < a12

for t = a

1 for t > a

g(t) G(s) = L[g(t)]

11

s

t1

s2

tν , ν > −1ν!

sν+1

e−αt 1

s + α

sin ωtω

s2 + ω2

cos ωts

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 . . .

JUNE 2008 MATH2019 Page 3

FOURIER SERIES

If f(x) has period p = 2L, then

f(x) = a0 +∞∑

n=1

(an cos

(nπ

Lx)

+ bn sin(nπ

Lx))

where

a0 =1

2L

∫ L

−L

f(x)dx

an =1

L

∫ L

−L

f(x) cos(nπ

Lx)

dx

bn =1

L

∫ L

−L

f(x) sin(nπ

Lx)

dx

LEIBNIZ’ THEOREM

d

dx

∫ v

u

f(x, t)dt =

∫ v

u

∂f

∂xdt + f(x, v)

dv

dx− f(x, u)

du

dx

Please see over . . .

JUNE 2008 MATH2019 Page 4

SOME BASIC INTEGRALS∫xndx =

xn+1

n + 1+ C for n 6= −1∫

1

xdx = ln |x|+ C∫

ekxdx =ekx

k+ C∫

axdx =1

ln aax + C for a 6= 1∫

sin kx dx = −cos kx

k+ C∫

cos kx dx =sin kx

k+ C∫

sec2 kx dx =tan kx

k+ C∫

cosec2kx dx = −1

kcot kx + C∫

tan kx dx =ln | sec kx|

k+ C∫

sec kx dx =1

k(ln | sec kx + tan kx|) + C∫

1

a2 + x2dx =

1

atan−1

(x

a

)+ C∫

1√a2 − x2

dx = sin−1(x

a

)+ C∫

1√x2 + a2

dx = sinh−1(x

a

)+ C∫

1√x2 − a2

dx = cosh−1(x

a

)+ C∫ π

2

0

sinn x dx =n− 1

n

∫ π2

0

sinn−2 x dx∫ π2

0

cosn x dx =n− 1

n

∫ π2

0

cosn−2 x dx

Please see over . . .

JUNE 2008 MATH2019 Page 5

Answer question 1 in a separate book

1. a) The matrix A is given by

A =

2 −2 2−2 1 12 1 1

.

i) Show that the vector

v =

2−11

is an eigenvector of the matrix A and find the corresponding eigen-value.

ii) Given that the other two eigenvalues of A are 2 and −2, find theeigenvectors corresponding to these two eigenvalues.

b) Given the integral ∫ 2

0

∫ 4

x2

x3√x4 + y2

dy dx.

i) Make a clear sketch of the region of integration.

ii) Express the integral with the order of integration reversed.

iii) Evaluate the integral you found in (ii).

Please see over . . .

JUNE 2008 MATH2019 Page 6

Answer question 2 in a separate book

2. a) Find

i) L(te−t sin 3t).

ii) L−1

{s + 1

s2 + 4s + 5+

e−2s

3s4

}.

b) The function f(t) is defined for t ≥ 0 by

f(t) =

1 0 ≤ t ≤ 1

t− 2 1 < t ≤ 2

0 t > 2.

i) Express f(t) in terms of the Heaviside function and hence or other-wise find L(f(t)), the Laplace transform of f(t).

c) Use the Laplace Transform method to solve the differential equation

y′′ − 4y′ + 4y = e2t , t > 0

subject to the initial condition y(0) = 1, y′(0) = 0.

Answer question 3 in a separate book

3. The odd periodic function f(x) is defined by

f(x) =

{−4− x −4 ≤ x ≤ 0

4− x 0 < x < 4(1)

with f(x + 8) = f(x) for all x.

a) Sketch f(x) for −12 ≤ x ≤ 8.

b) Find the coefficients in the Fourier series for the function defined byequation (1) and write out the series, explicitly giving the first threenon-zero terms in the series.

c) Find a particular solution of the ordinary differential equation

2d2y

dx2+ 11y = f(x)

in terms of a Fourier series where f(x) is given by equation (1) above.Hence, find the term(s) of largest magnitude in the particular solutionand give evidence to justify your answer.

Please see over . . .

JUNE 2008 MATH2019 Page 7

Answer question 4 in a separate book

4. The steady-state distribution of heat in a slab of width L is given by

∂2U

∂x2+

∂2U

∂y2= 0, for 0 < x < L, 0 < y < ∞

U(0, y) = U(L, y) = 0, for 0 < y < ∞U bounded as y → +∞U(x, 0) = f(x), for 0 ≤ x ≤ L.

a) Draw a clear diagram of the slab indicating the temperature U(x, y) onthree sides of the slab.

b) Use the method of separation of variables to find the general solutionU(x, y), where any unknown constants are related to U(x, 0) = f(x).You must explicitly consider all possibilities for the separation constantin your working.