MORE IN Applied Mathematics - 3
MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
May 2015
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Determine the constants a,b,c,d if f(z)=x2+2axy+by2+i(dx2+2cky+y2) is analytic.
5 M
1 (b) Find a cosine series of period 2π to represent sin x in 0≤x≤π.
5 M
1 (c) Evaluate by using Laplace Transformation $\int^\infty_0 e^{-3x} t \cos t \ dt.$
5 M
1 (d) A vector field is given by $\overline {F} = (x^2 + xy^2)i + (y^2 + x^2 y)j. Show that F is irrotational and find its potential. Such that F=∇ϕ. 5 M 2 (a) Solve by using Laplace Transform. (D2+2D+5)y=e-t sin t, when y(0)=0, y(0)=1. 6 M 2 (b) Find the total work done in moving a particle in the force field. F=3xy i-5z j+10x k along x=t2+1, y=2t2, z=t3 from t=1 and t=2. 6 M 2 (c) Find the Fourier series of the function f(x)=e-x. 0 8 M 3 (a) Prove that \[ J_{1/2}(x) = \sqrt{ \dfrac {2} {\pi x } }\cdot \sin x$
6 M
3 (b) Verify Green's theorem in the plane for ∮(x2-y)dx+(2y2+x)dy Around the boundary of region defined by y=x2 and y=4.
6 M
3 (c) Find the Laplace transforms of the following. $i) \ e^{-t} \int^t_0 \dfrac {\sin u} {u} du \\ ii) \ t \sqrt{1+\sin t}$
8 M

4 (a) f f(x)=C1Q1(x) + C2Q2(x) + C3Q3(x)t where C1, C2, C3 constants and Q1, Q2, Q3 are orthonormal sets on (a,b), show that. $\int^b_a [f(x)]^2 dx = c^2_1 + c^2_2 + c^2_3.$
6 M
4 (b) If v=ex sin y, prove that v is a Harmonic function. Also find the corresponding harmonic conjugate function and analytic function.
6 M
4 (c) Find inverse Laplace transform of the following: $i) \ \dfrac {S^2} {(S^2 + a^2) (S^2+b^2)} \\ ii) \ \dfrac {S+2}{S^2 -4S+13}$
8 M

5 (a) Find the Fourier series if f(x)=|x|, -k
6 M
5 (b) Define solenoidal vector. Hence prove that $$\overline {F} = \dfrac{\overline{a}\times\overline{r}}{r^n}$$ is a solenoidal vector.
6 M
5 (c) Find the bilinear transformation under which 1, i, -1 from the z-plane are mapped onto 0, 1, ∞ of w-plane. Further show that under this transformation the unit circle in w-plane is mapped onto a straight line in the z-plane. Write the name of this line.
8 M

6 (a) Using Gauss's Divergence theorem $$\iint_s \ \overline {F} .d\overline {s}$$ where F2x2yi-y2j+4xz2 k and s is the region bounded by y2+z2=9 and x=2 in the first octant.
6 M
6 (b) Define billinear transformation, And prove that in a general, a bilinear transformation maps a circle into a circle.
6 M
6 (c) Prove that $\int xJ_{2/3} (x^{3/2})dx = - \dfrac {2}{3} x^{-1/2}J_{-1/3}(x^{3/2}) .$
8 M

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