## 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 + x^{2}-y^{2}+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)=x^{2}+2axy+by^{2}+i(cx^{2}+2dxy+y^{2}) is analytic.(5 marks)
**2 (a)** Find complex form of forurier series f(x)=e^{3x} in 0<x<3< a="">

</x<3<><>(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)=x^{2} -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=y^{2}

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<><>(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)