Applied Mathematics 3 - Dec 2013

Information Technology (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) $$Find \ L^{-1}\left\{\frac{e^{\frac{4-3}{s}}}{{\left(s+4\right)}^{\frac{5}{2}}}\right\}$$(5 marks) 1 (b) Find the constant a,b,c,d and e If
$$ f\left(z\right)=\left(ax^4+bx^2y^2+cy^4+dx^2-2y^2\right)+i\ \left(4x^3y-exy^3+4xy\right)\ $$ is analytic.(5 marks) 1 (c) Obtain half range Fourier cosine series for f(x)=sin x, x ∈ (0, π). (5 marks) 1 (d) If r and r have their usual meaning and a is constant vector, prove that
$$abla{}x\left[\frac{a\ x\\bar{r}}{r^n}\right]=\frac{\left(2-n\right)}{r^n}a+\frac{n\left(a\cdot{}\bar{r}\right)\bar{r}}{r^{n+2}}$$(5 marks) 2 (a) Find the analytic function f(c) =u+iv if 3u+2v=y²- x² + 16 xy.(6 marks) 2 (b) Find the z-transform of $$\left\{a^{\left\vert{}k\right\vert{}}\right\}$$ and hence find the z-transform of $$\left\{{\left(\frac{1}{2}\right)}^{\left\vert{}k\right\vert{}}\right\}$$(6 marks) 2 (c) Obtain Fourier series expansion for $$f\left(x\right)=\sqrt{1-\cos{x,\ }}\ x\in{}\left(0,\ 2\pi{}\right)$$ and hence deduce that $$\sum_{n=1}^{\infty{}}\frac{1}{4n^2-1}=\frac{1}{2}.$$(8 marks) 3 (a) $$\left(i\right)\ \ L^{-1}\left\{\frac{s}{{\left(2\ s+1\right)}^2}\right\}$$
$$\left(ii\right)\ \ L^{-1}\left\{\log{\begin{array}{l}\frac{s^2+a^2}{\sqrt{s+b}}\\\ \end{array}}\right\}$$
(6 marks) 3 (b) Find the orthogonal trajectories of the family of curves e^-x cos y+xy=? where ? is the real constant in xy - plane.(6 marks) 3 (c) Show that $$ \bar{F}=(y\ e^{xy}\cos{z)i+(x\ e^{xy}\cos{z)\ j-(e^{xy}\sin{z)\ k\ }}} $$ is irrotational and find the scalar potential for $$ \ \bar{F}\ and\ evaluate\ \int_c^{\ }\bar{F}\ \ $$ dr along the curve joining the points (0,0,0) and (-1,2,π).(8 marks) 4 (a) Evaluate by Green's theorem ∫ e^-x sin y dx+e^-x cos y dy where c is the rectangle whose vertices are (0,0) (π,0) (π, π/2) and (0, π/2) (6 marks) 4 (b) Find the half range sine series for the function.
$$f\left(x\right)=\frac{2\ k\ x}{l},\ \ 0\leq{}x\leq{}\frac{l}{2}$$
$$f\left(x\right)=\frac{2k}{l}\left(l-x\right),\ \frac{l}{2}\leq{}z\leq{}l$$(6 marks) 4 (c) Find the inverse z-transform of $$\frac{1}{\left(z-3\right)\left(z-2\right)}$$
(i) |z|<2
(ii) 2<|z|<3
(iii) |z|>3.(8 marks) 5 (a) Solve using Laplace transform.
$$\frac{d^2y}{dx^2}+4\frac{dy}{dx}+3y=e^{-x},\ y\left(0\right)=1,\ y^{'}\left(0\right)=1.$$(6 marks) 5 (b) Express f(x)= π/2 e^-x cos x for x>0 as Fourier sine integral and show that
$$ \int_0^{\infty{}}\frac{w^3\sin{wx}}{w^4+4}\ dw=\frac{\pi{}}{2}\ e^{-x}\cos{x.} $$(6 marks) 5 (c) $$ Evaluate\ \iint_s^{\ }F\cdot{}nds,\ where\ \bar{F}=xi-yi+\left(z^2-1\right)k\ \ $$ and s s=is the cylinder formed by the surface z=0, z=1, x²+y²=4, using the Gauss-Divergence theorem. (8 marks) 6 (a) Find the inverse Laplace transform by using convolution theorem.
$$L^{-1}\left\{\frac{s^2+2s+3}{\left(s^2+2s+5\right)\left(s^2+2s+2\right)}\right\}$$(6 marks) 6 (b) Find the directional derivative of ? = 4e^2x-y+z at the point (1, 1, -1) in the direction towards the point (-3,5,6).(6 marks) 6 (c) Find the image of the circle x²+y²=1, under the transformation $$w=\frac{5-4z}{4z-2}$$(8 marks)