Sathyabama University 2008 B.E Computer Science and Engineering Engineering Mathematics - II - Question Paper

SATHYABAMA UNIVERSITY

(Established under section 3 of UGC Act, 1956)

 

Course & Branch: B.E/ B. Tech CSE/ECE/EEE/CIVIL/MECH/

CHEM (Part Time)

Title of the paper: Engineering Mathematics - II

Semester: II Max. Marks: 80

Sub.Code: 6CPT0009 (2006/2007/2007 JAN) Time: 3 Hours

 

PART A (10 x 2 = 20)

Answer All the Questions

1. Write the Dirichlets conditions?

2. State the Eulers formula for the interval (c, c + 2l).

3. Form the partial differential equation by eliminating the arbitrary constants a and b from Z = (x² + a) (y² + b).

4. Find the completer integral of P² + q² = 1.

5. Write down the three possible solns of wave equation.

6. An insulated rod of length l has its ends A and B kept at a degree centigrade and b degree centigrade until steady state conditions prevail. The temperature at each end in suddenly reduced to zero degree centigrade and dept so. Write the boundary conditions.

7. Write down the three possible solns of two dimensional heat equation in polar co-ordinates.

8. A Semi circular at 0C on the bounding diameter and 100C on its circumference. Write the corresponding boundary condition.

9. Write change of scale property.

10. Find the fourier cosine trans form of

 

 

 

 

PART B (5 x 12 = 60)

Answer All the Questions

 

11. (a) Find the fourier series of f(x) = x + x² in (-p, p) of periodicity 2p. Hence deduce

(b) Expand f(x) = x sinx as a cosine series in 0 < x < p and show that

(or)

12. (a) Find the foureir series of f(x) of Period 4 given by

(b) Find the complex form of the fourier series of f(x) = e^x in -p < x < p.

 

13. (a) Solve p²(1 + x²)y = qx².

 

(b) Find the general soln of x(z² y²) p + y(x² z²)q = z(y² x²)

(or)

14. (a) Solve (D³ 2D²D¹)z = Sin (x + 2y) + 3x²y.

 

(b) Form the PDE by eliminating f from z = xy + f(x² + y² + z²).

 

15. A bar of length 20cm has its a and at 30C and 80C until steady-state conditions Prevail, the temperature at A is rexised to 40C and at the same instant that at B is lowered to 60C and temperature are maintained there after. Find the temperature at distance X form the end A at time t.

(or)

16. A tightly stretched string with fixed ends pts x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating giving each point a velocity lx(l x), find the displacement.

 

17. The temperature u is maintained at 0C along three edges of a square plate of length 100C cm and the fourth edge is maintained at a constant temperature u₀ until steady-state conditions prevail. Find an expression for the temperature u at any point (x, y) of the plate. Calculate the temperature at the centre of the plate.

(or)

18. In a semicircular plate of radius a with bounding diameter at 0C and the circumference at tC, show that the steady-state temperature distribution is gun by sin (2n 1)q.

 

19. Find the fourier transform of f(x) given by

and hence evaluate (i)

(ii)

(or)

20. (a) Find the foureir cosine transform of .

(b) Evaluate using transform methods.