Get premium membership and access revision papers, questions with answers as well as video lessons.

Set 208: Mathematics For Engineers Iv Question Paper

Set 208: Mathematics For Engineers Iv

Course:Bachelor Of Science

Institution: Kenyatta University question papers

Exam Year:2009

DATE: Monday 5th January, 2009
TIME: 11.00 a.m. – 1.00 p.m.

INSTRUCTIONS
Answer question Number ONE and ANY TWO.

QUESTION ONE – (30 MARKS)
(a)
If f(x) is defined in the interval [-Ll L]. State the fourier series corresponding to
f(x).

[5 marks]
(b)
Expand f(x) = x2 -? < x < ? in a fourier series.

[7 marks]
(c)
(i)
Define a Gamma function.

[3 marks]
(ii)
Evaluate ?? 5 -x
x e

dx

[3 marks]
0
(d)
Solve the partial differential equation
2
? z x2
?
y given that z(x, 0) = x2 and z(1, y) = cosy.
[6 marks]
?x y
?
(e)
Solve the differential equation at x = 0.2 y´ = 2x + y using the Runge-
Kutta’s third order method defined by
Yn+1 = yn + ?(K1 + 4K2 + K3) for
K1 = hf(xo, yo); K2 = hf (xo + ½h, yo + ½K1) and
K3 = hf (xo + 1, yo + K2) for y(o) = 0. Let h = 0.1

2
QUESTION TWO – (20 MARKS)
(a)
State the difference between odd and even functions.
[5 marks]
(b)
Find the fourier series for
?-? ?
x ? 0

f(x) ?
?

?
?

0 ?
1
1

and deduce that 1
2

?
?
?

...
..........

?
?
12
8

[8 marks]
9 25

(c)
State and proof Perseval’s Identity.

[7 marks]

QUESTION THREE – (20 MARKS)
(a)
Prove that n ?
1 ? n
n

[4 marks]
(b)
Evaluate the following using prove of @ above

(i)
1
?
-3
x

[4 marks]
2
2

(ii)
15
?

[4 marks]
2
3
(c)
(i)
Evaluate the integral ??
-y
y e

dy

[4 marks]
0
(ii)
Define the Beta function and hence compute
B(2.5, 1.5)

[4 marks]

QUESTION FOUR – (20 MARKS)
(a)
Classify the following partial differential Equation as Elliptic, Hyperbolic or
Parabolic.
2
u
?
k? u

(i)

?

[3 marks]
2
t
?
x
?
2
2
2
? y
a ? y

(ii)

?

[3 marks]
2
2
t
?
x
?
(b)
Show that u(x, t) = e-8t Sin2x is a solution to the BVP
2
2
? y
? u

? 2

2
2
t
?
x
?
(
u ? ,t )? 0

for u(0, t) =

[8 marks]
u?x, 0
? ? S
in2x.

3

(c)
Solve using separation of variables
2
2
? y 4? y

?

[6 marks]
2
2
t
?
x
?

QUESTION FIVE – (20 MARKS)
A fourth order Runge-Kutta’s formula for solving the differential equation is

y = yo + ? (K1 + 2K2 +2K3 + K4) where

K1 = hf (xo, yo)

K2 = hf(xo + ½h, yo + ½K1)

K3 = hf (xo + ½h, yo + ½K2)

K4 = hf (xo + h, yo + K3)

Use the above formula to solve the d.e.

y'' = x + y when x = 0.5

and y = 1 when x = 0. Use h = 0.1

[20 marks]

………………..

Set 208: Mathematics For Engineers Iv Question Paper

More Question Papers

Popular Exams