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THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS AND STATISTICS

NOVEMBER 2009

MATH2019ENGINEERING 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 CALCULATORS WITH AN AFFIXED "UNSW APPROVED" STICKERMAY 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.

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NOVEMBER 2009 MATH2019 Page 2

TABLE OF LAPLACE TRANSFORMS AND THEOREMS

g(t) is a function defined for all t 2: 0, and whose Laplace transform

exists. The Heaviside step function u is defined to be

fort<afor t = a

fort> a

9(1) G(s) ~ £[9(1)]

11 -

s

1I -

s'

tV,v>-lv!--8 11+1

e-at 1--s +"

wsin wt

8 2 +w2

scos wt

82 + w2

u(1 - a)e-as

-s

I'(t) sF(s) - 1(0)

1"(1) s'F(s) - sl(O) - 1'(0)

e-o' l(t) F(s +,,)

1(1 - a)u(t - a) e-asF(s)

tl(t) -F'(s)

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NOVEMBER 2009 MATH2019

FOURIER SERIES

Page 3

1f f(x) has period p ~ 2L, then

where

VARIATION OF PARAMETERS

Suppose that Yh(X) = AY1(X) + BY2(X) is the general solution of the homoge­neous differential equation

y" + p(x)y' + q(x)y ~ 0,

where A and B are constants. Then a particular solution of the associatednon-homogeneous equation

y" + p(x)y' + q(x)y ~ f(x)

is given by

(x) ~ - (x) JY2(x)f(x) dx + y ( ) JYl(xJf(x) dYP Yl w(x) 2 X w(x) x

where w(x) ~ det (t;i:\ t;I:D ~ Yl(X)Y;(x) - Y2(X)Y;(x)

Please see over ...

NOVEMBER 2009 MATH2019

SOME BASIC INTEGRALS

JXn+l

xndx=--+C forn=l--ln+l{J;dx ~ In Ixl + 0

J ek"ekxdx = T +C

JaXdX=I~aax+C forai-l

J coskxsin kxdx = --k- +0

J sinkxcos kx dx = -k- + 0

J tankxsec2 kxdx= k +C

Jcoseczkxdx = -~ cotkx + C

J dInlseckxl 0

tankx X= k +

Jsec kx dx ~ ~(In I seckx + tan kxl) + 0

J '). 1 'ldx= ~tan-l ("'-) +0a +x a a

J 1 dx,=sin-1 ("'-) +0va2 x2 a

J 1 dx ~ sinh-1("'-) + 0

vx2+a2 a

J 1 dx=cosh-1 ("'-)+0vxz a2 a

l ' n-l1'sinnxdx = -- sinn - 2 xdxono

l ' n-l1'cosn xdx = -- cosn-2 xdxono

Page 4

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NOVEMBER 2009 MATH2019 Page 5

Answer question 1 in a separate book

1. a) The matrix A is given by

i) Show that the vector

is an eigenvector of the matrix A and find the corresponding eigen­value.

ii) Given that the other two eigenvalues of A are 2 and 4, find theeigenvectors corresponding to these two eigenvalues.

b) Consider the double integral

l'l' 2..;X eX' dx dyo y'

i) Sketch the region of integration.

ii) Evaluate the double integral with the order of integration reversed.

c) Use the method of variation of parameters to find the general solution ofthe differential equation

y" - 2y' + y = 35x3j2e':I:.

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NOVEMBER 2009 MATH2019 Page 6

Answer question 2 in a separate book

2. a) Find

i) £:(4t'e').

ii) £-1 {:2-:~}'b) The function 9(t) is given by

9(t)~{:, forO<:t<1fort~1.

i) Sketch the function 9(t) for 0 <: t <: 2.ii) Write g(t) in terms of the Heaviside step function.

iii) Hence, or otherwise, find the Laplace transform of g(t).

c) Use the Laplace transform method to solve the initial value problem

y" + 2y' - 3y ~ 6e-" with y(O) ~ 2 and y'(O) ~ -14.

Answer question 3 in a separate book

3. a) Find all critical points of the function

f(x,y) ~ 2x3 + 3x'y+y'-y

and classify each one as a relative maximum, relative minimum or saddlepoint.

b) Let f(x) ~ Ixl for x E [","I with f(x + 2,,) ~ f(x) for all x.

i) Make a sketch of this function for -31f ::; X :; 31T.

ii) Is f(x) odd, even or neither?

iii) Find the Fourier series of f(x}.iv) By considering the value at x = 1r in your answer for the Fourier

series in iii), find the value of

00 1

L: (2k + 1)2·k=O

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NOVEMBER 2009 MATH2019 Page 7

for 0:::; x :::; 11", t 2: 0,

Answer question 4 in a separate book

4. The vibrations of an elastic string of length 11" satisfy the wave equation

a2u {)2 U

at2 = ox2 '

where u(x, t) is the deflection of the string. The string is fixed at the endsx = 0 and x = 7r. Hence,

U(O, t) ~ °and U(K, t) ~ 0, for all t.

a) Assuming a solution of the form u(x, t) ~ F(x)G(t) show that

pll - kF = 0 and G" ~ kG = 0,

for some constant k.

b) Applying the boundary conditions (and considering all possibilities forthe constant k) show that

and that possible solutions for F(x) are

Fn(x) = sin(nx), for n = 1,2, ....

c) Find all possible solutions Gn(t) for G(t).

cl) Suppose now that the initial deflection and the velocity are given by

u(x,O) ~2sin(4x), for °<:: x <:: K, and

ou =0 when t = O.at 'Use'b) and your answer in c) to show that the solution is

u(x, t) ~ 2sin(4x) cos(4t).