MORE IN Applied Mathematics - 3
MU Civil Engineering (Semester 3)
Applied Mathematics - 3
December 2015
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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).
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.
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.
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
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

More question papers from Applied Mathematics - 3