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Posted By: Atul Member Level: Diamond Posted Date: 24 Apr 2008
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2006 Anna University B.E Civil CC 401 — MATHEMATICS — IV Question paper
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B.E. DEGREE EXAMINATION.
Fourth Semester
Civil Engineering and Computer Based Construction
CC 401 — MATHEMATICS — IV
Time : Three hours Maximum : 100 marks
Answer ALL questions.
PART A — (10 ´ 2 = 20 marks)
1. Write the Fourier series form of a function defined in an interval .
2. Classify odd or even function or neither of them.
(a)
(b) .
3. What is the order of the partial differential equation formed by eliminating the constants a, b and c from .
4. Solve .
5. Write any two variable separable solutions of .
6. Write wave equation and two dimensional steady state heat equation.
7. Under what assumptions one dimensional wave equation is derived in the form .
8. Write any two solutions of Laplace equation in polar form.
9. What is Rodrigue’s formula?
10. Write Bessel’s equation of order zero.
PART B — (5 ´ 16 = 80 marks)
11. (i) Solve the given differential equation interms of Bessel functions. (5)
.
(ii) Express interms of Legendre’s polynomials. (5)
(iii) Using the generating function prove that . (6)
12. (a) (i) Find a Fourier series to represent in .
Hence find the value of (8)
(ii) Find the Fourier series of (8)
Or
(b) (i) The turning moment T units of crank shaft of a steam engine is given for a set of crank angle .
: 0 30 60 90 120 150 180
T : 0 5224 8097 7850 5499 2626 0
Find the first three terms in a series of sines to represent T. (8)
(ii) Find the half range cosine series of in (o, l). Use those coefficients to find the sum of the series (8)
13. (a) (i) Form a partial differential equation p. d. e. by eliminating the arbitrary functions f and g from . (8)
(ii) Solve the p.d.e. . (8)
Or
(b) (i) Find all solutions of the p.d.e. . (8)
(ii) Solve . (8)
14. (a) (i) Find all variable separable solutions of . (6)
(ii) A tightly stretched string with fixed end points , is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points a velocity , find the displacement of the string at any distance x from one end and at any time t. (10)
Or
(b) (i) Find all solutions of one dimensional heat conduction equation. (6)
(ii) An insulated rod of length L has its ends A and B maintained at
0° and 100° respectively until steady state conditions prevail. If A is suddenly increased to 20° and B is reduced to 70° and maintained thereafter, find the subsequent temperature distribution. (10)
15. (a) (i) Find all solutions of Laplace equation in Cartesian co–ordinates. (6)
(ii) The diameter of a semicircular plate of radius ‘a’ is kept at 0°C and the temperature at the semicircular boundary is T°C. Find the steady state temperature in the plate. (10)
Or
(b) A square thin metal plate of side a is bounded by the lines . The edges x = 0, y = a are kept at zero temperature. The edge y = 0 is insulated and the edge x = a is kept at constant temperature . Find the steady state temperature at any point in the plate. (16)
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