APPLIED MATHEMATICS-3(paper no. 3) by SURESH KUMAR.....sureshkumarthegreat@yahoo.com

APPLIED MATHEMATICS-3

PAPER NO. 3

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                                               AM-201
                              APPLIED MATHEMATICS-III
                                  (B.Tech 3rd Semester,5001)
Time : 3 Hours                                                                                        Maximum Marks : 60
NOTE:- This paper consist of Three Sections. Section A is compulsory. Do any Four questions from
                 Section B and any two questions from Section C

                                     Section-A                                         Marks : 20

1(a) Find the locus represented by | z - 2 i | = 2.
(b) If f(z) = x² + i y², find the points in the z plane where f'(z) is defined. Also find its value at these points.
(c) Distinguish between the zeros and poles of a function w = f(z). Can an analytic function have zeros and poles ?
(d) Define J_n(x) and write the differential equation which has J_n(x) as its solution. What are the values of J_o(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 equation.
(g) Define Fourier sine-cosine over the interval -π to π. Is it possible to write this series for the constant f(x) = 2 over
    this interval.
(h) Define Laplace transform. If f(s) is the Laplace transform of f(t) then what is the laplace transform of

(i) Write the partial differeantial equation 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(θ). Also write its boundary
    conditions and initial conditions if any.

                                              Section-B                                          Marks:5 Each

2. Derive necessary form of C.R. equations for a function w = f(z) to be analytic.
    Also find the image of the circle | z - 1 | = 1 in w plane under the mapping w = z².
3. Show that with usual notions xⁿJ_n(x) is the solution of

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

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

    in the range 0 < x < π.
6. (a) Find the Laplace transform of
            f(t) = t/T, 0 < t <=T
                 = 1 , t > T
    (b) Find the inverse Laplace transform of (s + 2)/(s-2)³

                                               Section-C                                        Marks : 10 Each

7. (a) Evaluate

where c is | z - 2 | = 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 equation :
            (D² + 5D + 6)x = t e^t given that x=2, dx/dt = 1, at t = 0.
9. Use method of seperation of variables to solve the wave equation and use this solution to obtain the diflection
    u(x,t) of a vibrating string of length l whose end points are fixed and the string is given zero initial velocity
    and initial deflection:
            f(x) = 2kx/l; 0 < x < l/2
                   = (2k/l)(l-x); l/2 < x < l.

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