STUPIDSID
1 (a)
Find the Laplace transform of te-4 cosh2t.
5 M
1 (b)
Find the fixed points of \( w= \dfrac {3z-4}{z-1}. \) Also express it in the normal form \( \dfrac {1}{w-\alpha} = \dfrac {1} {z-\alpha} + \lambda\) where λ is a constant and α is the fixed point. Is this transformation parabolic?
5 M
1 (c)
Evaluate \( \int^{1+i}_0 (X^2 - iy)dz \) along the path
i) y=x ii) y=x^2
i) y=x ii) y=x^2
5 M
1 (d)
Prove that \[ f_i(x)=1, f_2(x)=x, f_3(x) = \dfrac {3x^2 -1} {2} \] are orthogonal over (-1, 1).
5 M
2 (a)
Find inverse Laplace transform of \( \dfrac {2s} {s^4+4} .\)
6 M
2 (b)
Find the image of the triangular region whose vertices are i, 1+i, 1-i under the transformation w=z+4-2i. Draw the sketch.
6 M
2 (c)
Obtain fourier expansion of f(x)=|cosx|in (-π,π).
8 M
3 (a)
Obtain complex form of fourier series for f(x)=cosh 2x+sinh2x in (-2,2).
6 M
3 (b)
Using Crank Nicholson simplified formula solve \( \dfrac { \partial^2 u} {\partial x^2} - \dfrac {\partial u} {\partial t} = 0 \) given \( u(0,t)=0, \ u(4,t)=0, \ u(x,0)= \dfrac {x} {3} (16-x^2) \) find uij for i=0, 1, 2, 3, 4 and j=0, 1, 2.
6 M
3 (c)
Solve the equation \( y + \int^1_0 ydt = 1-e-1. \)
8 M
4 (a)
Evaluate \( \int^{2\pi} {0} \dfrac {d \theta } {5+3\sin \theta } \)
6 M
4 (b)
Find half-range cosine series for f(x)=ex, 0
6 M
4 (c)
Obtain two district Laurent's series for \( f(z) = \dfrac {2z-3} {z^2 - 4z - 3} \) in powers of (z-4) indicating the regions of convergence.
8 M
5 (a)
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). Assume h=1 and find the values of u upto t=5.
6 M
5 (b)
Find the Laplace transform of \(e^{-4t} \int^1_0 u\sin 3 \ udu. \)
6 M
6 (a)
Find inverse Laplace transform of \( \dfrac {s} {(s^2 -a^2)^2 \) by using convolution theorem.
6 M
6 (b)
Find an analytic function f(z)=u+iv where u+v=ex(cosy + siny).
6 M
6 (c)
Solve the equation \( \dfrac {\partial u} {\partial t} = k \dfrac {\partial ^2 u} {\partial ^2 x^2} \) for the conduction of heat along a rod of length l subject to following conditions.
i) u is not infinity for t→∞
ii) \( \dfrac {\partial u} {\partial x} = 0 \) for x=0 and x=l for any time t
iii) u=lx-x^2 for t=0 between x=0 and x=l.
i) u is not infinity for t→∞
ii) \( \dfrac {\partial u} {\partial x} = 0 \) for x=0 and x=l for any time t
iii) u=lx-x^2 for t=0 between x=0 and x=l.
8 M
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