## Applied Mathematics - 3 - Dec 2016

### Mechanical Engineering (Semester 3)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1(a)** Evaluate $$\int _c(\bar{z}+2z)dz $$ along the circle x^{2+y2=1}(5 marks)
**1(b)** Evaluate the integral using Laplace Transform $$\int ^\infty _0 e^{-t}\left ( t\sqrt{1+\sin t} \right )dt$$(5 marks)
**1(c)** Determine the analytic function whose real part is u = r^{3} sin 30.(5 marks)
**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 marks)
**2(a)** Find complex from of Fourier series of f(x)=e^{2x} in (0,2)(6 marks)
**2(b)** Find the orthogonal trajectory of the family of curves given by 2x-x^{3}+3xy^{2}=a(6 marks)
**2(c)** Using Bender Schmidt method solve

$$\frac{\partial^2u }{\partial x^2}-\frac{\partial u}{\partial t}$$ = 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 marks)
**3(a)** Find k such that $$\frac{1}{2}\log \left ( x^2+y^2 \right )+i\tan^{-1}\left ( \frac{kx}{y} \right )$$ is analytic(6 marks)
**3(b)** Evaluate $$\int \frac{1}{\left ( z^3-1 \right )^2}$$dz where C is the circle |z-1|=1(6 marks)
**3(c)** Show that the set of function

$$\left \{ Sin\left ( \frac{\pi x}{2L} \right ),Sin\left ( \frac{3\pi x}{2L} \right ),Sin\left ( \frac{5\pi x}{2L} \right )...... \right \}$$ form an orthogonal set over the interval [0, L]. Construct corresponding orthonormal set.(8 marks)
**4(a)** Find Laplace Transform of the periodic function

$$\begin{Bmatrix}
sin2t,0<1 &\frac{\pi }{2} \\
\\0,\frac{\pi }{2}\ltt &="" \lt\pi="" \end{bmatrix}f(t)="\left" (="" t+\pi="" \right="" )$$<="" a="">

</t>(6 marks)
**4(b)** Find half range sine series for x sin x in (o,π)(6 marks)
**4(c)** Expand

$$f(z)=\frac{z^2-1}{z^2+5z+6}$$ around z=1(8 marks)
**5(a)** Using residue theorem evaluate $$\oint _c\frac{e}{\left ( z^2+\pi ^2 \right )^2}dz$$ where C is |z|=4(6 marks)
**5(b)** Find Fourier expansion of f(x)=x+x^{2} in (-π,

π) and f(x+2π)=f(x)(6 marks)
**5(c)** Find $ 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 marks)
**6(a)** Show that the function $w=\frac{4}{z} $/ transform the straight lines x=c in the z-plane into circles in the W-plane.(6 marks)
**6(b)** Solve using Laplace Transform$ R\frac{dQ}{dt}+\frac{Q}{C}=V $/, Q=0 when t=0(6 marks)
**6(c)** Solve the Laplace equation $\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 marks)