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SE-EXTC_SEM3 M3 DEC17

[Time: 3 Hours] $\hspace{50mm}$ [Total Marks: 80]

Please check whether you have got the right question paper

N.B:
1. Question No. 1 is compulsory 2. Attempt any Three (03) Questions from remaining Five (05) Questions.

Q1

A) Find Laplace transform of e$^{-4t}$ $\int_{0}^{t}{u sin3u du}$. $\hspace{20mm}$ 5

B) Find the orthogonal trajectories of the curves e$^{-x}$cos y+xy=$\alpha$, where $\alpha$ is a real constant in XY plane. $\hspace{20mm}$ 5

C) Find a Fourier series to represent f(x) = x$^{2}$ in (0, 2$\pi$) hence deduce that

$\frac{\pi ^{2}}{12}=\frac{1}{1}-\frac{1}{4}+\frac{1}{9}-\frac{1}{16}+ \ldots \ldots \ldots \ldots$ $\hspace{50mm}$ 5

D) Prove that $\overrightarrow{F}$ = (x$^{2}$ + xy$^{2}$) î +(y$^{2}$ +x$^{2}$y) j is irrotational and find its scalar potential. $\hspace{30mm}$5

Q2

A) If u = -r$^{3}$sin3$\theta$ , find analytic function whose real part is u. $\hspace{10mm}$6

B) Find the Bilinear transformation which maps the points z= 1, i, -1 onto the points w = i, 0, -i. $\hspace{20mm}$ 6

C) Obtain the Fourier series for $f(x) =\bigg\{ \begin{gathered} -\pi , -\pi \lt x \lt 0 \\ x , 0 \lt x \lt \pi \\ \end{gathered}$

Hence deduce that $\frac{\pi ^{2}}{8}=\frac{1}{1}+\frac{1}{9}+\frac{1}{25}+ \ldots \ldots \ldots \ldots$ $\hspace{40mm}$ 8

Q3

A) Find inverse Laplace transform of (i) tan$^{-1}$ $(\frac{2}{S})$ (ii) e$^{-4s}$ . $\frac{S}{(S+4)^{3}}$.$\hspace{20mm}$ 6

B) Find Complex form of Fourier Series of coshax + sinhax in (-a, a). $\hspace{20mm}$6

C) Verify Greens Theorem for $\int_{C}$ (xy + y$^{2}$)dx +x$^{2}$ dy where C is the closed curve of the region bounded by y=x and y=x$^{2}$. 8

Q4

A) Prove that $\int{x^{4}J_{1}(x) dx= x^{4}J_{2}(x)- 2x^{3}J_{3}(x)}$. $\hspace{25mm}$6

B) Use Gauss's Divergence theorem to evaluate, $ \iint_{S}{\overrightarrow{N }}. \overrightarrow{F} ds$ where $ \overrightarrow{F}$ = 4xi +3yj -2zk and S is the surface bounded by x=0, y=0, z=0 and 2x +2y+z=4. $\hspace{100mm}$6

C) Solve using Laplace transform( D$^{2}$ +2D+1)y =3te$^{-t}$ , given y(0)=4 and y'(0)=2. $\hspace{20mm}$8

Q5

A) Find half range cosine series for $f(x)=\bigg\{ \begin{gathered} x , 0 \lt x \lt \frac{\pi }{2} \\ \pi -x , \frac{\pi }{2} \lt x \lt \pi \\ \end{gathered}$. $\hspace{55mm}$6

B) Find the image of real axis in z-plane onto w-plane under the bilinear transformation $w=\frac{1}{z+i}$. $\hspace{25mm}$6

C) Prove that y = $\sqrt[]{x}$ . J$_{n}$(x) is a solution of the equation, $X^{2}\frac{d^{2}y}{dx^{2}}+(x^{2}-n^{2}+\frac{1}{4})y=0$. $\hspace{35mm}$8

Q.6

A) Find the constant a, b, c if the normal to the surface ax$^{2}$ +yz +bxz$^{3 }$= C at P(1, 2, 1) parallel to the surface y$^{2}$ +xz = 61 at (10,1,6). $\hspace{65mm}$6

B) Find inverse Laplace transform using convolution theorem $ \frac{S}{(S^{2}+9)^{2}}$ . $\hspace{45mm}$ 6

C) Express the function f(x) = $\bigg\{ \begin{gathered} 1 , |x|\lt1 \\ 0 , \vert x\vert \gt 1 \\ \end{gathered} $ as Fourier integral. Hence evaluate

$\int_{0}^{\infty }{\frac{sinw . sinwx}{w} dw}$. $\hspace{75mm}$ 8

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