Punjab Technical University 2007 Bachelor of Computer Science (B Level) APPLIED MATHEMATICS-III - Question Paper

APPLIED MATHEMATICS-III
(B.Tech third Semester,5001)
Time: 03 Hours Max marks: 60
Instruction to candidates:
4) Section –A is compulsory.
5) Attempt any 4 ques. from section-B
6) Attempt any 2 ques. from section-C



part -A
Q1) (marks 10*2=20)


1(a) obtain the locus represented by | z - two i | = 2.
(b) If f(z) = x2 + i y2, obtain the points in the z plane where f'(z) is described. Also obtain its value at these points.
(c) Distinguish ranging from the zeros and poles of a function w = f(z). Can an analytic function have zeros and poles ?
(d) describe Jn(x) and write the differential formula which has Jn(x) as its solution. elaborate the values of Jo(x) and J1(x) ?
(e) State Rodrigue's formula and use it to evaluate P2(x).
(f) Eliminate arbitary functions f and g from u = f(x + i y) + g(x - iy) and classify the resulting partial differential formula.
(g) describe Fourier sine-cosine over the interval -p to p. Is it possible to write this series for the constant f(x) = two over
this interval.
(h) describe Laplace transform. If f(s) is the Laplace transform of f(t) then what is the laplace transform of


(i) Write the partial differeantial formula which governs the steady state distribution of temperature inside a circular plate
whose both faces are insulated and the circumference is kept at steady temperature f(?). Also write its boundary
conditions and initial conditions if any.

Section-B Marks:5 every

2. Derive necessary form of C.R. equations for a function w = f(z) to be analytic.
Also obtain the image of the circle | z - one | = one in w plane under the mapping w = z2.
3. Show that with usual notions xnJn(x) is the solution of


4. Solve the subsequent partial differential equations:
(a) (y + z) p + (z + x) q = x + y
(b)


5. provided that c is a constant, show that it is possible to write:

in the range 0 < x < p.
6. (a) obtain the Laplace transform of
f(t) = t/T, 0 < t <=T
= one , t > T
(b) obtain the inverse Laplace transform of (s + 2)/(s-2)3

Section-C Marks : 10 every


7. (a) Evaluate


where c is | z - two | = 2.
(b) Use method of contour integration to evaluate:


8. (a) Prove that with usual notations:


(b) Use method of Laplace transform to solve the differential formula :
(D2 + 5D + 6)x = t et provided that x=2, dx/dt = 1, at t = 0.
9. Use method of seperation of variables to solve the wave formula and use this solution to find the diflection
u(x,t) of a vibrating string of length l whose end points are fixed and the string is provided zero initial velocity
and initial deflection:
f(x) = 2kx/l; 0 < x < l/2
= (2k/l)(l-x); l/2 < x < l.

Earning:   Approval pending.