Applied Mathematics - 3 - May 2016

Electronics 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 ^{\infty}_0e^{-2t}\left ( \dfrac{\sinh t \sin t}{t} \right )dt $(5 marks) 1(b) Obtain the Fourier Series expression for
f(x) = 9-x² in (-3,3)(5 marks) 1(c) Find the value of 'p' sucj that the function f(z) expressed in polar co-ordinates as
f(z) = r³ cos p&theta(5 marks) 1(d) If $ \bar{F}=(y^2-z^2+3yz-2x)\hat{i}+(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}. $
Show that $ \bar{F} $ is irrotational and solenoidal.(5 marks) 2(a) Solve the differential equation using Laplace Transform $ \dfrac{d^2y}{dt^2}+4\frac{dy}{dt}+8y=1 $, given y(0)=0 and y'(0)=1(6 marks) 2(b) Prove that $$J_4(x)=\left ( \dfrac{48}{x^3}-\dfrac{8}{x} \right )J_1(x)-\left ( \dfrac{24}{x^2}-1 \right )J_0(x)$$(6 marks) 2(c)(i) Find the directional derivative of ϕ = 4xz³ - 3x²y²z at (2,-1,2) in the direction of $ 2\hat{i}+3\hat{j}+6\hat{k}. $(8 marks) 2(c)(ii) if $ \bar{r}=x\hat{i}+y\hat{j}+z\hat{k} $
prove that $ \nabla \log r=\dfrac{\bar{r}}{r^2} $(8 marks) 3(a) Show that { cosx, cos2x, cos3x.....} is a set of orthogonal fundtions over (-π, π), Hence construct an orthogonal set.(6 marks) 3(b) Find an analytic function f(z)=u+iy where. $$u=\dfrac{x}{2}\log(x^2+y^2)-y\tan^{-1}\left ( \dfrac{y}{x} \right )+\sin x \cosh y$$(6 marks)

Find the Laplace transform of

3(c)(i) $ \int ^1_0 ue^{-3u}\cos^2u du $(4 marks) 3(c)(ii) $ t\sqrt{1+\sin t} $(4 marks) 4(a) Find the Fourier Series for $ f(x)=\dfrac{3x^2-6\pi x+2\pi^2}{12} $ in (0,2π)
Hence deduce that $ \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}\cdots=\dfrac{\pi^2}{6} $(6 marks) 4(b) Prove that $$\int ^b_0 xJ_0(ax)dx=\dfrac{b}{a}J_1(ab)$$(6 marks)

Find

4(c)(i) $ L^{-1}\left [ \log\left ( \dfrac{s^2+1}{s(s+1)} \right ) \right ] $(4 marks) 4(c)(ii) $ L^{-1}\left [ \log\left ( \dfrac{s+2}{s^2-2s+17} \right ) \right ] $(4 marks) 5(a) Obtain the half range cosine series for $$\begin {align*} f(x)&=x, 0\ltx\lt\frac{\pi}{2}\ \="" \\="" &="\pi-x," \frac{\pi}{2}\ltx\lt\pi="" \end="" {align*}$$<="" a="">

</x<\frac{\pi}{2}>(6 marks) 5(b) Find the Bi-linear Transformation which maps the points 1,i,-1 of z plane onto i,0,-i of w-plane(6 marks) 5(c) Verify Green's Theorem for $ \int _c \overline{F} .\overline{br} $ where $ \bar{F}=(x^2-xy)\hat{i}+(x^2-y^2)\hat{j} $ and C is the curve bounded by x² = 2y and x=y(8 marks) 6(a) Show that the transformation $ w=\dfrac{i-iz}{1+z} $ maps the unit circle |z|=1 into real axis of w plane.(6 marks) 6(b) Using Convolution theorem find $$L^{-1}\left [ \dfrac{s}{(s^2+1)(s^2+4)} \right ]$$(6 marks) 6(c)(i) Use Gauss Divergence Theorem to evaluate $ \iint _s \bar{F}.\hat{n}ds\ \text{where}\ \bar{F}=x\hat{i}+y\hat{j}+z\hat{k} $ and S is the sphere x² + y² + z² = 9 and $ \hat{n} $ is the outward normal to S(8 marks) 6(c)(ii) Use Stoke's Theorem to evaluate $ \int _c\overline{F}.\overline{dr}\ \text{where}\ \bar{F}=x^2\hat{i}-xy\hat{j} $ and C is the square in the plane z=0 and bounded by x=0, y=0, x=a and y=a.(8 marks)