## 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 t^{3} 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.<><>(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 (D^{2}+2D+5)y=e^{-t}sint when y(0)=0 and y'(0)=1.(8 marks)
**4 (a)** Evaluate ? f(z)dz along the Parabola y=2x^{2} from z=0 to z=3+18i where f(z)=x^{2}-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)