MORE IN Applied Mathematics - 3
MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2014
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) Prove that f(z)= x2-y2+2ixy is analytic and find f'(z).
5 M
1 (b) Find the Fourier series expansion for f(x)=|x|, in (-?, ?).
5 M
1 (c) Using Laplace transform solve the following differential equation with given condition $\dfrac {d^2y}{dt^2} + y=t,$ given that y(0)=1 & y'(0)=0.
5 M
1 (d) $if \ \bar{A} =\nabla (xy + yz + zx), \ find \ \nabla \cdot \bar{A} \ and \ \nabla \time \bar{A}$
5 M

2 (a) $if \ L [j_0 (t) ] = \dfrac {1}{\sqrt{s^2+1}}$ prove that $\int^\infty_0 e^{-6 t} t j_0 (4 t) dt=3/500. 6 M 2 (b) Find the directional derivative of ?=x4+y4+z4 at A(1, -2, 1) in the direction of AB where B is (2,6,-1). Also find the maximum directional derivative of ? at (1, -2, 1). 6 M 2 (c) Find the Fourier series expansion for f(x)=4-x2, in (0,2). Hence deduce that \[ \dfrac {\pi^2}{6} = \dfrac {1}{1^2} + \dfrac {1}{2^2} + \dfrac {1}{3^2} + \cdots \ \cdots$
8 M

3 (a) Prove that $J_{1/2} (x) = \sqrt{\dfrac {2}{\pi x}} \sin x$
6 M
3 (b) Using Green's theorem evaluate $\int_C (2x^2-y^2) dx + (x^2 + y^2) dy$ where 'C' is the boundary of the surface enclosed by the lines x=0, y=0, x=2, y=2.
6 M
3 (c) (i) Find Laplace Transform of $e^{-3t} \int^1_0 u \sin 3u \ du$
4 M
3 (c) (ii) Find the Laplace transform of $\dfrac {d}{dt} \left ( \dfrac {1- \cos 2t}{t} \right )$
4 M

4 (a) Obtain complex form of Fourier series for the functions f(x)=sin ax in (-?, ?), where a is not an integer.
6 M
4 (b) Find the analytic function whose imaginary part is $v=\dfrac {x}{x^2+y^2} + cosh \ y\cdot \cos x$
6 M
4 (c) Find inverse Laplace Transform of following $i) \ \log \left ( \dfrac {s^2 + a^2}{\sqrt{s+b}} \right ) \\ ii) \ \dfrac {1} {s^3 (s-1)}$
8 M

5 (a) Obtain half-range cosine series for f(x)=x(2-x) in 0 < x< 2.
6 M
5 (b) Prove that $\overline {F} = \dfrac {\overline r} {r^3}$ is both irrotational and solenoidal.
6 M
5 (c) Show that the function u=sin x cosh y+2 cos x sinh y+x2-y2+4xy satisfies Laplace's equation and find it corresponding analytic function.
8 M

6 (a) Evaluate by Stroke's theorem $\int_c (x,y, \ dx + x \ y^2 \ dy )$ where C is the square in the xy-plane with vertices (1,0), (0,1),(-1,0) and (0,-1).
6 M
6 (b) Find the bilinear transformation, which maps the points z=-1, 1, &infity; onto the points w=-i, -1, i.
6 M
6 (c) Show that the general solution of $\dfrac {d^2y}{d x^2} + 4x^2 y =0 \ is \ y =\sqrt{x} [A \ J_{1/4} (x^2) + B \ J_{-1/4} (x^2) ]$ where A and B are constants.
8 M

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