Applied Mathematics - 3 - Dec 2014

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 t³ cost.(5 marks) 1 (b) Find the image of |z-ai|=a under the transformation w=1/z.(5 marks) 1 (c) Construct an analytic function, whose real part is 2^2x (x cos 2y-y sin 2y).(5 marks) 1 (d) Show that the set of functions cos nx n=1,2,3,... is orthogonal on (0,2?).(5 marks) 2 (a) By using Convolution Theorem. Find inverse Laplace transform pf $$ \dfrac {1}{s^2(s+1)^2 }. $$(6 marks) 2 (b) Find bilinear transformation that maps the points 2,i,-2 onto the point 1,i,-1.(6 marks) 2 (c) Find Fourier Series for f(x)=cos mx in (?, &-pi;) where m is not an integer. Deduce that $$ \cos m\pi = \dfrac {2m}{\pi} \left (\dfrac {1}{2m^2}+ \dfrac {1}{2m^2-1^2} + \dfrac {1}{m^2-2^2} \cdots \ \cdots \dfrac {1}{m^2-n^2} \right ) $$ hence show that $$ \sum^\infty _1 \dfrac {1}{9n^2-1}= \dfrac {1}{2} - \dfrac {\pi \sqrt{3}}{18}$$(8 marks) 3 (a) Find complex form Fourier series f(x)-e^3x in 0<x<3.< a="">

</x<3.<></span>(6 marks) 3 (b) Using Crank Nicholson method solve $$ \dfrac {\partial ^2 u} {\partial x^2} = \dfrac {\partial u} {\partial t} $$ subject to 0 ≤ x ≤ 1 u(0,t)=0 u (1,t)=0, u(x,0)=100x(1-x) taking h=0.25 in one step.(6 marks) 3 (c) Using Laplace solve (D²+2D+5)y=e^-tsint when y(0)=0 and y'(0)=1.(8 marks) 4 (a) Evaluate ? f(z)dz along the Parabola y=2x² from z=0 to z=3+18i where f(z)=x²-2iy.(6 marks) 4 (b) Find half range cosine series for $$ \begin{align*} f(x) &= x & 0 < x \pi /2 \ \ \ \ \\ &= \pi -x & \ \pi /2 < x < \pi \end{align*} $$(6 marks) 4 (c) Obtain two distinct Laurent's series of $$ f(z) = \dfrac {1}{(1+z^2)(z+2)} \ for \ 1<|z|<2 \ and \ |z|>2. $$(8 marks) 5 (a) By using Bender Schmitt method solve $$ \dfrac {\partial^2 f}{\partial x^2} = \dfrac {\partial f}{\partial t} f(0,t) = f (5, t)=0. \ f(x,0)=x^2 (25-x^2) $$ find f in range taking h=1 and up to 5 seconds.(6 marks) 5 (b) Evaluate $$ \int^\infty_0 e^{-t} \dfrac {\sin^2 t }{t} dt . $$(6 marks) 5 (c) Evaluate $$ \int^{2\pi}_0 \dfrac {\cos 3 \theta } {5-4 \cos \theta } d \theta $$(8 marks) 6 (a) A string is stretched and fastened to two points distance l apart, motion is started by displacing the string in the form $$ y =a \sin \left ( \dfrac {\pi x}{l} \right ) $$ from which it is released at time t=0. Show that the displacement of a point at a distance x from on end at a distance x from one end at time t is given by $$ y(x,t)- a \sin \left ( \dfrac {\pi x}{l} \right ) \cos \left ( \pi \dfrac {ct}{l} \right ) $$(6 marks) 6 (b) If f(z)=u+iv is analytic and u-v=e^x(cos y-sin y) find f(z) in terms of z.(6 marks) 6 (c) Evaluate: $$ L^{-1} \left [\dfrac {s}{(s-2)^6} \right ] $$(8 marks)