B.E./B.Tech. DEGREE EXAMINATION, april/MAY 2003.
Third Semester
ma 231 — mathematics – III
(Common to All Branches)
Time : Three hours Maximum : 100 marks
Answer ALL the questions.
PART A — (10 ´ 2 = 20 marks)
1. Find the complete integral of 
where 
and 
.
where and .2. Solve 
3. If 
is expressed as a Fourier series in the interval (–2, 2), to which value this series converges at x = 2?
is expressed as a Fourier series in the interval (–2, 2), to which value this series converges at x = 2?4. If the Fourier series corresponding to 
= x in the interval 
is 
without finding the values of 
find the value of 
= x in the interval is without finding the values of find the value of 5. Classify the following second order partial differential equations :
(a) 
(b) 
6. Write any two solutions of the Laplace equation obtained by the method of separation of variables.
7. Obtain the Laplace transform of 
in the simplified form.
in the simplified form.8. Find 
9. If F(s) is the Fourier transform of , write the formula for the Fourier transform of 
in terms of F.
, write the formula for the Fourier transform of in terms of F.10. State the convolution theorem for Fourier transforms.
PART B — (5 ´ 16 = 80 marks)
11. (i) Prove that the Laplace transform of the triangular wave of period 
defined by
defined by
is 
.
. (ii) Solve using Laplace transforms :
12. (a) (i) Solve : 
(ii) Solve : 
Or
(b) (i) Form the partial differential equation by eliminating the arbitrary functions f and g in 
(ii) Solve : 
13. (a) (i) Obtain the Fourier series for 
in 
. Deduce that 
in . Deduce that (ii) Obtain the constant term and the first harmonic in the Fourier series expansion for where is given in the following table :
where is given in the following table : x : | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| : | 18.0 | 18.7 | 17.6 | 15.0 | 11.6 | 8.3 | 6.0 | 5.3 | 6.4 | 9.0 | 12.4 | 15.7 |
Or
(b) (i) Expand the function 
as a Fourier series in the interval 
.
as a Fourier series in the interval . (ii) Obtain the half range cosine series for 
in the interval 
Deduce that 
in the interval Deduce that 14. (a) A tightly stretched flexible string has its ends fixed at x = 0 and x = l. At time t = 0, the string is given a shape defined by 
where k is a constant, and then released from rest. Find the displacement of any point x of the string at any time t > 0.
where k is a constant, and then released from rest. Find the displacement of any point x of the string at any time t > 0.Or
(b) The ends A and B of a rod l cm long have the temperatures 40°C and 90°C until steady state prevails. The temperature at A is suddenly raised to 90°C and at the same time that at B is lowered to 40°C. Find the temperature distribution in the rod at time t. Also show that the temperature at the mid point of the rod remains unaltered for all time, regardless of the material of the rod.
15. (a) (i) Find the Fourier transform of 
if a > 0. Deduce that 
if a > 0.
if a > 0. Deduce that if a > 0. (ii) Find the Fourier sine transform of 
Or
(b) (i) Find the Fourier cosine transform of
Hence prove that 
(ii) Derive the Parseval's identity for Fourier transforms.
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