MU Applied Mathematics - 3 - December 2013 Exam Question Paper

MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
December 2013

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 of sin √t

5 M

1 (b) Show that the set of functions

sin (\frac{π x}{2 L}), sin (\frac{3 π x}{2 L}), sin (\frac{5 π x}{2 L})

$\sin \left ( \dfrac {\pi x}{2L} \right ), \sin \left (\dfrac {3 \pi x}{2L} \right ), \sin \left ( \dfrac {5\pi x}{2L} \right )$ 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)=e^3x in 0

6 M

2 (b) Using Crank Nicholson Method solve u_t-u_xx 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

$(i) \ \dfrac {\pi^{2}}{6}=\sum^{\infty}_{1}\dfrac {1}{n^{2}}\\(ii)\ \dfrac {\pi^{2}}{12}=\sum^{\infty}_{1}\dfrac {(-1)^{n+1}}{n^{2}}\\(iii)\ \dfrac {\pi^{2}}{8}= \dfrac {1}{1^{2}}+\dfrac {1}{3^{2}}+\dfrac {1}{5^{2}}+....$

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

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.

6 M

5 (b) Find half range Cosine Series for f(x)=e^x 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}}\$

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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