Applied Mathematics - 3 - Dec 2014

Electronics Engineering (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Prove that f(z)= x²-y²+2ixy is analytic and find f'(z).(5 marks) 1 (b) Find the Fourier series expansion for f(x)=|x|, in (-?, ?).(5 marks) 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 marks) 1 (d) $$ if \ \bar{A} =\nabla (xy + yz + zx), \ find \ \nabla \cdot \bar{A} \ and \ \nabla \time \bar{A} $$(5 marks) 2 (a) $$ if \ L [j_0 (t) ] = \dfrac {1}{\sqrt{s^2+1}}$$ prove that $$ \int^\infty_0 e^{-6 t} t j_0 (4 t) dt=3/500.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Find the directional derivative of ?=x\ltsup\gt4\lt/sup\gt+y\ltsup\gt4\lt/sup\gt+z\ltsup\gt4\lt/sup\gt at A(1, -2, 1) in the direction of AB where B is (2,6,-1). Also find the maximum directional derivative of ? at (1, -2, 1).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c)\lt/b\gt Find the Fourier series expansion for f(x)=4-x\ltsup\gt2\lt/sup\gt, in (0,2). Hence deduce that $$ \dfrac {\pi^2}{6} = \dfrac {1}{1^2} + \dfrac {1}{2^2} + \dfrac {1}{3^2} + \cdots \ \cdots $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a)\lt/b\gt Prove that $$ J_{1/2} (x) = \sqrt{\dfrac {2}{\pi x}} \sin x $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt Using Green's theorem evaluate $$ \int_C (2x^2-y^2) dx + (x^2 + y^2) dy $$ where 'C' is the boundary of the surface enclosed by the lines x=0, y=0, x=2, y=2.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c) (i)\lt/b\gt Find Laplace Transform of $$ e^{-3t} \int^1_0 u \sin 3u \ du$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c) (ii)\lt/b\gt Find the Laplace transform of $$ \dfrac {d}{dt} \left ( \dfrac {1- \cos 2t}{t} \right ) $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a)\lt/b\gt Obtain complex form of Fourier series for the functions f(x)=sin ax in (-?, ?), where a is not an integer.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Find the analytic function whose imaginary part is $$ v=\dfrac {x}{x^2+y^2} + cosh \ y\cdot \cos x $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt Find inverse Laplace Transform of following $$ i) \ \log \left ( \dfrac {s^2 + a^2}{\sqrt{s+b}} \right ) \ ii) \ \dfrac {1} {s^3 (s-1)} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Obtain half-range cosine series for f(x)=x(2-x) in 0 < x< 2.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Prove that $$ \overline {F} = \dfrac {\overline r} {r^3} $$ is both irrotational and solenoidal.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt Show that the function u=sin x cosh y+2 cos x sinh y+x\ltsup\gt2\lt/sup\gt-y\ltsup\gt2\lt/sup\gt+4xy satisfies Laplace's equation and find it corresponding analytic function.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt Evaluate by Stroke's theorem $$ \int_c (x,y, \ dx + x \ y^2 \ dy ) $$ where C is the square in the xy-plane with vertices (1,0), (0,1),(-1,0) and (0,-1).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Find the bilinear transformation, which maps the points z=-1, 1, &infity; onto the points w=-i, -1, i.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt Show that the general solution of $$ \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 marks)