Applied Mathematics - 3 - Dec 2015

Mechanical 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) Find Laplace transform of tsin³t.(5 marks) 1 (b) Find half range sine series in (0, π) for x(π-x).(5 marks) 1 (c) Find the image of the rectangular region bounded by
x=0, x=3, y=0, y=2 under the transformation ω=z+(1+i).(5 marks) 1 (d) Evaluate ∫f(z)dz along the parabola y=2x², z=0 to z=3+18i where f(z)=x²-2iy.(5 marks) 2 (a) Find two Laurent's series of $$ f(z) = \dfrac {1}{z^3 (z-1)(z+2)} $$ about z=0 for
i) |z|<1     ii) 1<|z|<2.(8 marks) 2 (b) Find complex form of Fourier series for f(x)=cos h2x+sin h2x in (-2,2).(6 marks) 2 (c) Find bilinear transformation that maps 0,1,&infty; of the z plane into -5, -1, 3 of ω plane.(6 marks) 3 (a) Solve by using Laplace transformation
(D² + 2D+5) y=e^-1 sint when y(0) and y¹(0)=1.(8 marks) 3 (b) Solve $$ \dfrac {\partial^2 u}{\partial x^2} - 2 \dfrac {\partial u}{\partial t} =0 $$ by Bender Schmidt method given u(0,t)=0, u(4,t)=0, u(x,0)-x(4-x)(6 marks) 3 (c) Expand f(x)=lx-x² 0<x<1 in="" a="" half="" range="" cosine="" series.<="">

</x<1>(6 marks) 4 (a) Evaluate $$ \int^{2\pi}_{0} \dfrac {2\theta}{(2+cos \theta)^2}\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Evaluate $$ \int^{\infty}_0 e^{-2t} \dfrac {\cos 2t \sin 3t}{t}dt $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt Using Crank Nicholson method solve. $$ \dfrac {\partial ^2 u}{\partial x^2}- \dfrac {\partial u}{\partial t} =0 \ u(0,t)=0, \ u(4,t)=0 \ u(x,0) = \dfrac {x}{3} (16-x^2) $$ Find u\ltsub\gt8\lt/sub\gt for i=0,1,2,3,4 and j=0,1,2.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Find analytic function whose real part is $$ \dfrac{\sin 2x}{\cosh 2y+\cos 2x} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Find $$ i) \ L^{-1}\left [ \dfrac {e^{-\pi 3}}{s^2 -2s+2} \right ] \ ii) \ L^{-1}\left [ \tan^{-1} \left ( \dfrac {s+a}{b} \right ) \right ] $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt Find the solution of one dimensional heat equation $$ \dfrac {\partial u}{\partial t} =e^2 \dfrac {\partial ^2}{\partial x^2} $$ under the boundary conditions u(0, t)=0 \ltbr\gt u(l,t)=0 and u(x,0)=x \ltbr\gt 0\ltx\ltl, l="" being="" length="" of="" the="" rod.\lt="" a=""\gt\ltbr\gt\ltbr\gt \lt/x\ltl,\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt A string is stretched and fastened to two points distance l apart. Motion is started by displacing the string in the form y=a sin (πx/l) which it is released at time t=0. Show that the displacement of a point at a distance x from one end at time t is given by $$ y_{x, t}=a \sin \left ( \dfrac {\pi x}{l} \right ) \cos \left ( \dfrac {\pi ct} { l}\right ) $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Find the residue of $$ \dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)^2} $$at its poles.(6 marks) 6 (c) Find Fourier series of xcosx in (-π, π).(6 marks)