Applied Mathematics - 3 - Dec 2013

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 of sin √t(5 marks) 1 (b) Show that the set of functions$$ \sin \left ( \dfrac {\pi x}{2L} \right ), \sin \left (\dfrac {3 \pi x}{2L} \right ), \sin \left ( \dfrac {5\pi x}{2L} \right ) $$ is orthogonal over (O,L).(5 marks) 1 (c) Show that u=sinx cos hy +2 cos x sin hy + x²-y²+4xy Statifies laplace equation and find its corresponding analytic function f(z)=u+iv(5 marks) 1 (d) Determine constant a,b,c,d if f(z)=x²+2axy+by²+i(cx²+2dxy+y²) is analytic.(5 marks) 2 (a) Find complex form of forurier series f(x)=e^3x in 0<x<3< a="">

</x<3<></span>(6 marks) 2 (b) Using Crank Nicholson Method solve u_t-u_xx subject to u(x,0)=0 u(0,t)=0 and u(1,t)=t for two time steps.(6 marks) 2 (c) Solve using laplace transforms $$ \dfrac {d^{2}y}{dt^{2}}+y=t, y(0)=1, y'(0)=0 $$ (8 marks) 3 (a) Find bilinear transformation that maps the points 0,1 -∞ of the z plane into -5,-1,3 of w plane.(6 marks) 3 (b) By using Convolution Theorem find inverse laplace transform of $$ \dfrac {1}{(S^{2}+4S+13)^{2}} $$(6 marks) 3 (c) Find fourier series of f(x)=x² -π ≤ x≤π and prove that
$$ (i) \ \dfrac {\pi^{2}}{6}=\sum^{\infty}_{1}\dfrac {1}{n^{2}}\$$ii)\ \dfrac {\pi^{2}}{12}=\sum^{\infty}_{1}\dfrac {(-1)^{n+1}}{n^{2}}\$$iii)\ \dfrac {\pi^{2}}{8}= \dfrac {1}{1^{2}}+\dfrac {1}{3^{2}}+\dfrac {1}{5^{2}}+....$$(8 marks) 4 (a) $$ Evaluate \ \int^{\infty}_{0}e^{-t} \dfrac {\sin^{2}t}{t}dt $$(6 marks) 4 (b) $$ Solve \ \dfrac {\partial^{2}u}{\partial x^{2}}-32 \dfrac {\partial u}{\partial t}=0 \ by$$
Bender schmidt method subject to conditions u(0,t)=0 u(x,0)=0 u(l,t)=t taking h=0.25 0< x <1(6 marks) 4 (c) Obtain two distinct Laurent's Serier for $$ f(z)= \dfrac {2z-3}{Z^{2}-4z-3} $$in powers of (z-4) indicating Region of Convergence.(8 marks) 5 (a) Evaluate $$ \int^{1+i}_{0} Z^2 dz $$ along
(i) line y=x
(ii) Parabola x=y²
is line independent of path? Eplain.(6 marks) 5 (b) Find half range Cosine Series for f(x)=e^x 0<x<1< a="">

</x<1<></span>(6 marks) 5 (c) Find analytic function f(z) =u+iv such that $$ u-v=\dfrac {\cos x +\sin x -e^{-y}}{2\cos x -e^y -e^{-y}}\\ when \ f\left ( \dfrac {pi }{2} \right )=0 $$ (8 marks) 6 (a) A tightly streached sting with fixed end points x=0 x=l in the shape defined by y=Kx(l-x) where K is a constant is released from this position of rest. Find y(x,t) the vertical displacement,
$$ if \ \dfrac {\partial^{2}y}{\partial t^{2}}=C^{2}\dfrac {\partial^{2}y}{\partial x^{2}}\ $$(6 marks) 6 (b) Find image of region bounded by x=0, x=2 y=0 y=2 in the z-plane under the transformation w=(1+j)Z(6 marks) 6 (c) $$ Evaluate \ \int^{2 \pi}_{0}\dfrac {d\theta}{25-16 \cos^{2} \theta}$$(8 marks)