Applied Mathematics - 3 - May 2015

Electronics & Telecomm. (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) Determine the constants a,b,c,d if f(z)=x²+2axy+by²+i(dx²+2cky+y²) is analytic. (5 marks)

1 (b) Find a cosine series of period 2π to represent sin x in 0≤x≤π. (5 marks)

1 (c) Evaluate by using Laplace Transformation $$ \int^\infty_0 e^{-3x} t \cos t \ dt. $$ (5 marks)

1 (d) A vector field is given by $$ \overline {F} = (x^2 + xy^2)i + (y^2 + x^2 y)j. Show that \ltspan style="text-decoration: overline"\gtF\lt/span\gt is irrotational and find its potential. Such that \ltspan style="text-decoration: overline"\gtF\lt/span\gt=∇ϕ. \lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 2 (a) \lt/b\gt Solve by using Laplace Transform. \ltbr\gt (D\ltsup\gt2\lt/sup\gt+2D+5)y=e\ltsup\gt-t\lt/sup\gt sin t, when y(0)=0, y(0)=1. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 2 (b) \lt/b\gt Find the total work done in moving a particle in the force field. \ltspan style="text-decoration: overline"\gtF\lt/span\gt=3xy i-5z j+10x k along x=t\ltsup\gt2\lt/sup\gt+1, y=2t\ltsup\gt2\lt/sup\gt, z=t\ltsup\gt3\lt/sup\gt from t=1 and t=2. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 2 (c) \lt/b\gt Find the Fourier series of the function f(x)=e\ltsup\gt-x\lt/sup\gt. 0\ltx\lt2π and="" f(x+2π)="f(x)." hence="" deduce="" that="" the="" value="" of="" $$="" \sum^\infty="" _{n="2}" \dfrac="" {(-1)^n}{n^2+1^n}.="" <="" a="">

</x<2π></span>(8 marks)

3 (a) Prove that $$ J_{1/2}(x) = \sqrt{ \dfrac {2} {\pi x } }\cdot \sin x $$ (6 marks)

3 (b) Verify Green's theorem in the plane for ∮(x²-y)dx+(2y²+x)dy Around the boundary of region defined by y=x² and y=4. (6 marks)

3 (c) Find the Laplace transforms of the following. $$ i) \ e^{-t} \int^t_0 \dfrac {\sin u} {u} du \\ ii) \ t \sqrt{1+\sin t} $$ (8 marks)

4 (a) f f(x)=C₁Q₁(x) + C₂Q₂(x) + C₃Q₃(x)_t where C₁, C₂, C₃ constants and Q₁, Q₂, Q₃ are orthonormal sets on (a,b), show that. $$ \int^b_a [f(x)]^2 dx = c^2_1 + c^2_2 + c^2_3. $$ (6 marks)

4 (b) If v=e^x sin y, prove that v is a Harmonic function. Also find the corresponding harmonic conjugate function and analytic function. (6 marks)

4 (c) Find inverse Laplace transform of the following: $$ i) \ \dfrac {S^2} {(S^2 + a^2) (S^2+b^2)} \\ ii) \ \dfrac {S+2}{S^2 -4S+13} $$ (8 marks)

5 (a) Find the Fourier series if f(x)=|x|, -k<x<k. hence="" deduce="" that="" $$="" \sum\dfrac="" {1}{(2n-1)^4}="\dfrac{\pi^4}{96}." \lt="" a=""\gt\ltbr\gt\ltbr\gt \lt/x\ltk.\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 5 (b) \lt/b\gt Define solenoidal vector. Hence prove that \ltspan class="math-text"\gt $$ \overline {F} = \dfrac{\overline{a}\times\overline{r}}{r^n} $$ \lt/span\gt is a solenoidal vector. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 5 (c) \lt/b\gt Find the bilinear transformation under which 1, i, -1 from the z-plane are mapped onto 0, 1, ∞ of w-plane. Further show that under this transformation the unit circle in w-plane is mapped onto a straight line in the z-plane. Write the name of this line. \lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 6 (a) \lt/b\gt Using Gauss's Divergence theorem \ltspan class="math-text"\gt $$ \iint_s \ \overline {F} .d\overline {s} $$ \lt/span\gt where \ltspan style="text-decoration: overline"\gtF\lt/span\gt2x\ltsup\gt2\lt/sup\gtyi-y\ltsup\gt2\lt/sup\gtj+4xz\ltsup\gt2\lt/sup\gt k and s is the region bounded by y\ltsup\gt2\lt/sup\gt+z\ltsup\gt2\lt/sup\gt=9 and x=2 in the first octant. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 6 (b) \lt/b\gt Define billinear transformation, And prove that in a general, a bilinear transformation maps a circle into a circle. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt 6 (c) \lt/b\gt Prove that $$ \int xJ_{2/3} (x^{3/2})dx = - \dfrac {2}{3} x^{-1/2}J_{-1/3}(x^{3/2}) .$$ </span>(8 marks)