MU Applied Mathematics - 3 - December 2016 Exam Question Paper

MU Mechanical Engineering (Semester 3)
Applied Mathematics - 3
December 2016

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) Evaluate

\int_{c} (\bar{z} + 2 z) d z

$\int _c(\bar{z}+2z)dz$ along the circle x^2+y²=1

5 M

1(b) Evaluate the integral using Laplace Transform

\int_{0}^{\infty} e^{- t} (t \sqrt{1 + \sin t}) d t

$\int ^\infty _0 e^{-t}\left ( t\sqrt{1+\sin t} \right )dt$

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)=e^2x 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^{2} u}{\partial x^{2}} - \frac{\partial u}{\partial t}

$\frac{\partial^2u }{\partial x^2}-\frac{\partial u}{\partial t}$ = 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 (x^{2} + y^{2}) + i \tan^{- 1} (\frac{k x}{y})

$\frac{1}{2}\log \left ( x^2+y^2 \right )+i\tan^{-1}\left ( \frac{kx}{y} \right )$ is analytic

6 M

3(b) Evaluate

\int \frac{1}{{(z^{3} - 1)}^{2}}

$\int \frac{1}{\left ( z^3-1 \right )^2}$ dz where C is the circle |z-1|=1

6 M

3(c) Show that the set of function

{S i n (\frac{π x}{2 L}), S i n (\frac{3 π x}{2 L}), S i n (\frac{5 π x}{2 L}) . . . . . .}

$\left \{ Sin\left ( \frac{\pi x}{2L} \right ),Sin\left ( \frac{3\pi x}{2L} \right ),Sin\left ( \frac{5\pi x}{2L} \right )...... \right \}$ 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}

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} + 5 z + 6}

$f(z)=\frac{z^2-1}{z^2+5z+6}$ around z=1

8 M

5(a) Using residue theorem evaluate

\oint_{c} \frac{e}{{(z^{2} + π^{2})}^{2}} d z

$\oint _c\frac{e}{\left ( z^2+\pi ^2 \right )^2}dz$ 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)

6 M

5(c) Find

i) L (e^{- 4 t} \int_{0}^{t} u \sin 3 u d u) i i) L^{- 1} (\frac{1}{s} l o g (1 + \frac{1}{s^{2}}))

$i)\ L\left ( e^{-4t}\int_{0}^{t} u\sin 3udu\right )\ \ ii) L^{-1}\left ( \frac{1}{s} log\left ( 1+\frac{1}{s^2} \right )\right )$

8 M

6(a) Show that the function

w = \frac{4}{z}

$w=\frac{4}{z}$ / 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{d Q}{d t} + \frac{Q}{C} = V

$R\frac{dQ}{dt}+\frac{Q}{C}=V$ /, Q=0 when t=0

6 M

6(c) Solve the Laplace equation

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0

$\frac{\partial^2u }{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$ / for the following data by sucessive interations (Calculate first two interations)
!mage

8 M

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