MERU UNIVERSITY OF SCIENCE AND TECHNOLOGY
P.O. Box 972-60200 – Meru-Kenya.
Tel: 020-2069349, 061-2309217. 064-30320 Cell phones: +254 712524293,
+254789151411, Fax: 064-30321
Website: www.must.ac.ke Email: info@must.ac.ke
University Examinations 2015/2016
THIRD YEAR, SECOND SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN MATHEMATICS AND COMPUTER SCIENCE, BACHELOR OF SCIENCE.
SMA 2320: ANALYTICAL APPLIED MATHEMATICS I
DATE: APRIL 2016
TIME: 2 HOURS
INSTRUCTIONS: Answer question one and any other two questions
QUESTION ONE (30 MARKS)
a) Find the laplace transform of each of the following functions
(i)
t t
(3 marks)
(ii)
2, 0  t  3
f t   
t , t  3
(4 marks)
b) Show that  u , v    v, u 
(5 marks)
c) Form a partial differential equation from each of the following function


(i)
z  f x2  y2
(ii)
2 x  ax  y   b
(3 marks)
2
(4 marks)
d) Find all the singulars points of the equation x 2  x  2  y 11  2 x  2  y 1   x  3 y  0 and
2
classify them as either regular or irregular.
e) Prove that J  n
x
  1 J n
n
x
Meru University of Science & Technology is ISO 9001:2008 Certified
Foundation of Innovations
(5 marks)
(6 marks)
Page 1
QUESTION TWO (20 MARKS)
1
a) Prove that    
2
b) Express J

3
2
(4 marks)
x  in terms of cosine and sine functions.
c) Solve the partial differential equation y 2 p  xyq  x z  2 y 
d) Find the inverse Laplace transform of the function
s2
s  8s  25
2
(5 marks)
(6 marks)
(5 marks)
QUESTION THREE (20 MARKS)
a) Find the Fourier series for the periodic function
0,
f x   
 x,
  x  0
(10 marks)
0 x


b) Solve the partial differential equation x 2  y 2  z 2 p  2 xyq  2 xz
(10 marks)
QUESTION FOUR (20 MARKS)
a) Using Laplace transforms, find the solution of the initial value problem
2
d2y
dy
 5  2 y  e  2t given that y 0   1, y 1 0   1
2
dt
dt
b) Find a power series expansion for the function f  x   x 2 in 0  x  2
Meru University of Science & Technology is ISO 9001:2008 Certified
Foundation of Innovations
(11 marks)
(9 marks)
Page 2