STUPIDSID
1(a)
Evaluate \[\int _c(\bar{z}+2z)dz \] along the circle x2+y2=1
5 M
1(b)
Evaluate the integral using Laplace Transform \[\int ^\infty _0 e^{-t}\left ( t\sqrt{1+\sin t} \right )dt\]
5 M
1(c)
Determine the analytic function whose real part is u = r3 sin 30.
5 M
1(d)
A rod length l has its ends A and B kept at 0°C and 100 respectively until steady state conditions prevail. If the tempreature at B is reduced sufddenly to 0°C and kept so while that of A is maintained. Find the tempreature u(x,t) at a distance from A and at time t.
5 M
2(a)
Find complex from of Fourier series of f(x)=e2x in (0,2)
6 M
2(b)
Find the orthogonal trajectory of the family of curves given by 2x-x3+3xy2=a
6 M
2(c)
Using Bender Schmidt method solve
\[\frac{\partial^2u }{\partial x^2}-\frac{\partial u}{\partial t}\] = 0 subject to the conditions u (o,t)=0,
u(1,t) =0,
u(x,0) = sinπx,
0≤x≤1. Assume h=0.2
\[\frac{\partial^2u }{\partial x^2}-\frac{\partial u}{\partial t}\] = 0 subject to the conditions u (o,t)=0,
u(1,t) =0,
u(x,0) = sinπx,
0≤x≤1. Assume h=0.2
8 M
3(a)
Find k such that \[\frac{1}{2}\log \left ( x^2+y^2 \right )+i\tan^{-1}\left ( \frac{kx}{y} \right )\] is analytic
6 M
3(b)
Evaluate \[\int \frac{1}{\left ( z^3-1 \right )^2}\]dz where C is the circle |z-1|=1
6 M
3(c)
Show that the set of function
\[\left \{ Sin\left ( \frac{\pi x}{2L} \right ),Sin\left ( \frac{3\pi x}{2L} \right ),Sin\left ( \frac{5\pi x}{2L} \right )...... \right \}\] form an orthogonal set over the interval [0, L]. Construct corresponding orthonormal set.
\[\left \{ Sin\left ( \frac{\pi x}{2L} \right ),Sin\left ( \frac{3\pi x}{2L} \right ),Sin\left ( \frac{5\pi x}{2L} \right )...... \right \}\] form an orthogonal set over the interval [0, L]. Construct corresponding orthonormal set.
8 M
4(a)
Find Laplace Transform of the periodic function
\[\begin{Bmatrix} sin2t,0<1 &\frac{\pi }{2} \\ \\0,\frac{\pi }{2}
\[\begin{Bmatrix} sin2t,0<1 &\frac{\pi }{2} \\ \\0,\frac{\pi }{2}
6 M
4(b)
Find half range sine series for x sin x in (o,π)
6 M
4(c)
Expand
\[f(z)=\frac{z^2-1}{z^2+5z+6}\] around z=1
\[f(z)=\frac{z^2-1}{z^2+5z+6}\] around z=1
8 M
5(a)
Using residue theorem evaluate \[\oint _c\frac{e}{\left ( z^2+\pi ^2 \right )^2}dz\] where C is |z|=4
6 M
5(b)
Find Fourier expansion of f(x)=x+x2 in (-π,
π) and f(x+2π)=f(x)
π) and f(x+2π)=f(x)
6 M
5(c)
Find \( i)\ L\left ( e^{-4t}\int_{0}^{t} u\sin 3udu\right )\ \ ii)
L^{-1}\left ( \frac{1}{s} log\left ( 1+\frac{1}{s^2} \right )\right ) \)
8 M
6(a)
Show that the function \(w=\frac{4}{z} \)/ transform the straight lines x=c in the z-plane into circles in the W-plane.
6 M
6(b)
Solve using Laplace Transform\( R\frac{dQ}{dt}+\frac{Q}{C}=V \)/, Q=0 when t=0
6 M
6(c)
Solve the Laplace equation \(\frac{\partial^2u }{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0 \)/ for the following data by sucessive interations (Calculate first two interations)
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8 M
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