STUPIDSID
1 (a)
Find laplace of sin √t
5 M
1 (b)
Show that the set of functions\[ \sin \left ( \dfrac {\pi x}{2L} \right ), \sin \left (\dfrac {3 \pi x}{2L} \right ), \sin \left ( \dfrac {5\pi x}{2L} \right ) \] is orthogonal over (O,L).
5 M
1 (c)
Show that u=sinx cos hy +2 cos x sin hy + x2-y2+4xy Statifies laplace equation and find its corresponding analytic function f(z)=u+iv
5 M
1 (d)
Determine constant a,b,c,d if f(z)=x2+2axy+by2+i(cx2+2dxy+y2) is analytic.
5 M
2 (a)
Find complex form of forurier series f(x)=e3x in 0
6 M
2 (b)
Using Crank Nicholson Method solve ut-uxx subject to u(x,0)=0 u(0,t)=0 and u(1,t)=t for two time steps.
6 M
2 (c)
Solve using laplace transforms \[ \dfrac {d^{2}y}{dt^{2}}+y=t, y(0)=1, y'(0)=0 \]
8 M
3 (a)
Find bilinear transformation that maps the points 0,1 -∞ of the z plane into -5,-1,3 of w plane.
6 M
3 (b)
By using Convolution Theorem find inverse laplace transform of \[ \dfrac {1}{(S^{2}+4S+13)^{2}} \]
6 M
3 (c)
Find fourier series of f(x)=x2 -π ≤ x≤π and prove that
\[ (i) \ \dfrac {\pi^{2}}{6}=\sum^{\infty}_{1}\dfrac {1}{n^{2}}\\(ii)\ \dfrac {\pi^{2}}{12}=\sum^{\infty}_{1}\dfrac {(-1)^{n+1}}{n^{2}}\\(iii)\ \dfrac {\pi^{2}}{8}= \dfrac {1}{1^{2}}+\dfrac {1}{3^{2}}+\dfrac {1}{5^{2}}+....\]
\[ (i) \ \dfrac {\pi^{2}}{6}=\sum^{\infty}_{1}\dfrac {1}{n^{2}}\\(ii)\ \dfrac {\pi^{2}}{12}=\sum^{\infty}_{1}\dfrac {(-1)^{n+1}}{n^{2}}\\(iii)\ \dfrac {\pi^{2}}{8}= \dfrac {1}{1^{2}}+\dfrac {1}{3^{2}}+\dfrac {1}{5^{2}}+....\]
8 M
4 (a)
\[ Evaluate \ \int^{\infty}_{0}e^{-t} \dfrac {\sin^{2}t}{t}dt \]
6 M
4 (b)
\[ Solve \ \dfrac {\partial^{2}u}{\partial x^{2}}-32 \dfrac {\partial u}{\partial t}=0 \ by\]
Bender schmidt method subject to conditions u(0,t)=0 u(x,0)=0 u(l,t)=t taking h=0.25 0< x <1
Bender schmidt method subject to conditions u(0,t)=0 u(x,0)=0 u(l,t)=t taking h=0.25 0< x <1
6 M
4 (c)
Obtain two distinct Laurent's Serier for \[ f(z)= \dfrac {2z-3}{Z^{2}-4z-3} \]in powers of (z-4) indicating Region of Convergence.
8 M
5 (a)
Evaluate \[ \int^{1+i}_{0} Z^2 dz \] along
(i) line y=x
(ii) Parabola x=y2
is line independent of path? Eplain.
(i) line y=x
(ii) Parabola x=y2
is line independent of path? Eplain.
6 M
5 (b)
Find half range Cosine Series for f(x)=ex 0
6 M
5 (c)
Find analytic function f(z) =u+iv such that \[ u-v=\dfrac {\cos x +\sin x -e^{-y}}{2\cos x -e^y -e^{-y}}\\ when \ f\left ( \dfrac {pi }{2} \right )=0 \]
8 M
6 (a)
A tightly streached sting with fixed end points x=0 x=l in the shape defined by y=Kx(l-x) where K is a constant is released from this position of rest. Find y(x,t) the vertical displacement,
\[ if \ \dfrac {\partial^{2}y}{\partial t^{2}}=C^{2}\dfrac {\partial^{2}y}{\partial x^{2}}\ \]
\[ if \ \dfrac {\partial^{2}y}{\partial t^{2}}=C^{2}\dfrac {\partial^{2}y}{\partial x^{2}}\ \]
6 M
6 (b)
Find image of region bounded by x=0, x=2 y=0 y=2 in the z-plane under the transformation w=(1+j)Z
6 M
6 (c)
\[ Evaluate \ \int^{2 \pi}_{0}\dfrac {d\theta}{25-16 \cos^{2} \theta}\]
8 M
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