MU Applied Mathematics - 3 - May 2015 Exam Question Paper

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MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
May 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 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

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)=e^x, 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=e^x(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.

8 M

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