Ecu 203:Engeneering Mathematics Viii Question Paper

Ecu 203:Engeneering Mathematics Viii

Course:Bachelor Of Science In Electrical And Electronic Engineering

Institution: Kenyatta University question papers

Exam Year:2012

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KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2011/2012
SECOND SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE
ECU 203: ENGINEERING MATHEMATICS VIII

DATE: FRIDAY 13TH APRIL 2012 TIME: 8.00 A.M. – 10.00 A.M.

Instructions: Answer any three questions

Question One

(a)
Find the Laplace transform of

(i)
3
4t
t ? e

(1 mark)

(ii)
sin 3t + cos 4t

(1 mark)

(b)
Find the Laplace inverse transform of

s ? 4

(i)

(3 marks)
2
s ? 4

a
a
a

(ii)
1
2
3
?
?

(2 marks)
2
3
s
s
s

6s ? 4

(iii)

(3 marks).
s2 ? s

(c)
Solve y ? ? 4 ?
y ? 4y ? ,
0 y?0? ? ,
0 y ??0? ? 2

(10 marks).

Question Two

(a)
The temperature u( ,
x t) in a semi- infinite rod 0 ? u ? ? is determined by the

differential equation.
2
u
?
? u

? k

2
t
?
x
?

And the conditions

(i)
u ? 0 when t ? ,
0 u ? 0

Page 1 of 2

(ii)
u ? u ? constant, when x ? 0 and t ? 0
0

Making use of sine transform, show that
?
2
sin x
? ?
2 ?

u?x,t?
? Kw t
? u
?
?
1? e
?dw
? 0
?
?
0
? ?
?

(20 marks)
Question Three
2
?
1 n cos nx
2
?? ?
a) (i)
Use Fourier series to show that x ?
? ??
4
(8 marks)
2
3
n?1
n
(ii)
Find the half cosine series of the function

?x 0 ? x ? ?

f (x)= ?

?0 ? ? ? x ? 0

(5 marks)
(b)
(i)
Find the Fourier integral representation of

?1 0 ? x ? 3

f (x)= ?

?0 x ? 3

(3 marks)
(ii)
Find the Fourier transform of the function
?k 0 ? x ? a

f (x) = ?

?0 a ? 0

(3 marks)

(iii) Find (5
? )

(3 marks)
2

Question Four
a)
Evaluate (i)
t
t * e

(3 marks)

(ii) 1*sint

(3 marks)
b)
Use the convolution theorem to evaluate
i.

(3 marks)
ii.

(3 marks)
c) Using the Laplace transformation, solve
i.

(4 marks)
ii.

(4 marks)

Question Five
Find the temperature u?x,t? in a laterally insulated copper bar length, L ? ? m if the initial
temperature f(x)= 0
10 C and the ends are kept at 0
0 C and 2
c ? 1.
(20 marks)

Page 2 of 2

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