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)=1z3(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 ∂2u∂x2−2∂u∂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 ∫∞0e−2tcos2tsin3ttdt
6 M
4 (c)
Using Crank Nicholson method solve. ∂2u∂x2−∂u∂t=0u(0,t)=0, u(4,t)=0u(x,0)=x3(16−x2) 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 sin2xcosh2y+cos2x
8 M
5 (b)
Find i) L−1[e−π3s2−2s+2]ii) L−1[tan−1(s+ab)]
6 M
5 (c)
Find the solution of one dimensional heat equation ∂u∂t=e2∂2∂x2 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 yx,t=asin(πxl)cos(πctl)
8 M
6 (b)
Find the residue of sinπz2+cosπz2(z−1)(z−2)2at its poles.
6 M
6 (c)
Find Fourier series of xcosx in (-π, π).
6 M
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