MORE IN Applied Mathematics - 3
MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2013
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) Prove that real and imaginary parts of an analytic function F(z)=u+iv are harmonic function.
5 M
1 (b) Find the fourier series for f(x)=|sin x| in (-Π,Π).
5 M
1 (c) Find the Laplace Transform of ?$\int_0^t\ {ue}^{-3u}\sin{4}udu$
5 M
1 (d) $if \ \bar{F}=xye^{2z}\widehat{i}+xy^2 \ \cos z \widehat{j}+x^2 \cos xy\ \widehat{k},\ find \ \bar{F} \ and \ curl \ \bar{F}$
5 M

2 (a) Using laplace transofrm solve-
(D2 + 3D + 2) y= e-2t. Sin t where y(0) = 0 and y' (0) =0.
6 M
2 (b) Find the directional derivative of d=x2 y cosz at (1,2, Π/2)in the direction of t=2i + 3j + 2k.
6 M
2 (c) Find the Fourier expansion of $f(x)=\sqrt{1-cosx}\ in\left(0,2\pi{}\right).Hence\ prove\ \frac{1}{2}=\sum_{n=1}^{\infty{}}\frac{1}{4n^2-1}.$
8 M

3 (a) $Prove \ the \ J_{3/2}(x)= \sqrt{\dfrac{2}{\pi x}} \left \{\dfrac {\sin x}{x}- \cos x \right \}$
6 M
3 (b) Evaluate by Green's theorem $\oint_c\left(x^2ydx+y^3dy\right)$ where C is closed path formed by y=x,y=x2
6 M
3 (c) i) Find the Laplace Transform of $\frac{\cos{bt-\cos{at}}}{t}$
ii) Find the Laplace Transform of $\frac{d}{dt}\left[\frac{sint}{t}\right].$
8 M

4 (a) Show that the set of functions {sinx, sin3x.....} OR {sin(2n+1)x:n=0,1,2,3....} is orthogonal over [0, π/2],Hence construct orthonormal set of functions.
6 M
4 (b) Find the imaginary part whose real part is u= x3 - 3xy2 + 3x2 + 1
6 M
4 (c) Find Inverse Laplace Transform of?
$i)log\left(\frac{s^2+4}{s^2+9}\right)$
$ii)\frac{s}{\left(s^2+4\right)\left(s^2+9\right)}$
8 M

5 (a) Obtain half range sine series for f(x)=x2 in 0
6 M
5 (b) A Vector field F is given by $\bar{F}=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}$ is irroational and hence find scalar point function ϕ such that F = Δ ϕ
6 M
5 (c) Show that the function V=ex (xsiny+ycosy) satisfies Laplace equation and find its corresponding analytic function
8 M

6 (a) By using stoke's theorem ,evaluate
$\oint_c\left[\left(x^2+y^2\right)\hat{i}+\left(x^2-y^2\right)\hat{j}\right]\cdot d\bar{r}$
where c is the boundary of a region enclosed by circles x2 + y2 =4, x2 + y2 = 16.
6 M
6 (b) Show that under the transformation w= 5-4z/4z-2 the circle |z|=1 in the z plane is transformed into a circle of unity in w-plane.
6 M
6 (c) Prove that $\int J_3\left(x\right)dx=\ -\frac{2J_1(x)}{x}-J_2(x)$
8 M

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