STUPIDSID
1 (a)
Find Laplace transform of tsin3t.
5 M
1 (b)
Find half range sine series in (0, π) for x(π-x).
5 M
1 (c)
Find the image of the rectangular region bounded by
x=0, x=3, y=0, y=2 under the transformation ω=z+(1+i).
x=0, x=3, y=0, y=2 under the transformation ω=z+(1+i).
5 M
1 (d)
Evaluate ∫f(z)dz along the parabola y=2x2, z=0 to z=3+18i where f(z)=x2-2iy.
5 M
2 (a)
Find two Laurent's series of \[ f(z) = \dfrac {1}{z^3 (z-1)(z+2)} \] about z=0 for
i) |z|<1 ii) 1<|z|<2.
i) |z|<1 ii) 1<|z|<2.
8 M
2 (b)
Find complex form of Fourier series for f(x)=cos h2x+sin h2x in (-2,2).
6 M
2 (c)
Find bilinear transformation that maps 0,1,&infty; of the z plane into -5, -1, 3 of ω plane.
6 M
3 (a)
Solve by using Laplace transformation
(D2 + 2D+5) y=e-1 sint when y(0) and y1(0)=1.
(D2 + 2D+5) y=e-1 sint when y(0) and y1(0)=1.
8 M
3 (b)
Solve \[ \dfrac {\partial^2 u}{\partial x^2} - 2 \dfrac {\partial u}{\partial t} =0 \] by Bender Schmidt method given u(0,t)=0, u(4,t)=0, u(x,0)-x(4-x)
6 M
3 (c)
Expand f(x)=lx-x2 0
6 M
4 (a)
Evaluate \[ \int^{2\pi}_{0} \dfrac {2\theta}{(2+cos \theta)^2}
8 M
4 (b)
Evaluate \[ \int^{\infty}_0 e^{-2t} \dfrac {\cos 2t \sin 3t}{t}dt \]
6 M
4 (c)
Using Crank Nicholson method solve. \[ \dfrac {\partial ^2 u}{\partial x^2}- \dfrac {\partial u}{\partial t} =0 \\ u(0,t)=0, \ u(4,t)=0 \\ u(x,0) = \dfrac {x}{3} (16-x^2)
\] Find u8 for i=0,1,2,3,4 and j=0,1,2.
6 M
5 (a)
Find analytic function whose real part is \[ \dfrac{\sin 2x}{\cosh 2y+\cos 2x} \]
8 M
5 (b)
Find \[ i) \ L^{-1}\left [ \dfrac {e^{-\pi 3}}{s^2 -2s+2} \right ] \\ ii) \ L^{-1}\left [ \tan^{-1} \left ( \dfrac {s+a}{b} \right ) \right ] \]
6 M
5 (c)
Find the solution of one dimensional heat equation \[ \dfrac {\partial u}{\partial t} =e^2 \dfrac {\partial ^2}{\partial x^2} \] under the boundary conditions u(0, t)=0
u(l,t)=0 and u(x,0)=x
0
u(l,t)=0 and u(x,0)=x
0
6 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 (πx/l) which it is released at time t=0. Show that the displacement of a point at a distance x from one end at time t is given by \[ y_{x, t}=a \sin \left ( \dfrac {\pi x}{l} \right ) \cos \left ( \dfrac {\pi ct} { l}\right ) \]
8 M
6 (b)
Find the residue of \[ \dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)^2} \]at its poles.
6 M
6 (c)
Find Fourier series of xcosx in (-π, π).
6 M
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