1 (a)
Find Laplace transform of t

^{3}cost.
5 M

1 (b)
Find the image of |z-ai|=a under the transformation w=1/z.

5 M

1 (c)
Construct an analytic function, whose real part is 2

^{2x}(x cos 2y-y sin 2y).
5 M

1 (d)
Show that the set of functions cos nx n=1,2,3,... is orthogonal on (0,2?).

5 M

2 (a)
By using Convolution Theorem. Find inverse Laplace transform pf \[ \dfrac {1}{s^2(s+1)^2 }. \]

6 M

2 (b)
Find bilinear transformation that maps the points 2,i,-2 onto the point 1,i,-1.

6 M

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 M

3 (a)
Find complex form Fourier series f(x)-e

^{3x}in 0
6 M

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 M

3 (c)
Using Laplace solve (D

^{2}+2D+5)y=e^{-t}sint when y(0)=0 and y'(0)=1.
8 M

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 M

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 M

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 M

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 M

5 (b)
Evaluate \[ \int^\infty_0 e^{-t} \dfrac {\sin^2 t }{t} dt . \]

6 M

5 (c)
Evaluate \[ \int^{2\pi}_0 \dfrac {\cos 3 \theta } {5-4 \cos \theta } d \theta \]

8 M

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 M

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 M

6 (c)
Evaluate: \[
L^{-1} \left [\dfrac {s}{(s-2)^6} \right ] \]

8 M

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