B.E. APPLIED MATHEMATICS- III-Mumbai-December 2019 - Grad Plus

# B.E. APPLIED MATHEMATICS- III-Mumbai-December 2019

## Semester: 3

[Total Time: 3hours]
[Total marks: 80]
N.B. 1) Question No. 1 is compulsory.
2) Answer any Three from remaining
3) Figures to the right indicate full marks.
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Q. 1) a) Find Laplace transform of ????(????) = ⅇ−4???? ???????????? 3????. ????????????2????.   (5M)

b) Show that the set of functions ????(????) = 1, ????(????) = ???? are orthogonal on (−1,1).
Determine the constants a and b such that the function ℎ(????) = −1 + ???????? + ????????2
is orthogonal to both ????(????) and ????(????) (5M)

c) Evaluate ∫c (????2 − 2????̅ ???? + 1)???????? where C is the circle |????| = 1.  (5M)

d) Compute the Spearman’s Rank correlation coefficient R and Karl Pearson’s correlation coefficient r from the following data. (5M)

 x 12 17 22 27 32 y 113 119 117 115 121

Q. 2) a) Using Laplace transform, evaluate. \int_0^\infty e^{{}^{-t}\int_0^t\frac{\sin\;u}udu\;dt.}. (6M)

b) Find an analytic function ????(????) = ???? + ????????, if. (6M)
???? = ⅇ−???? {(????2 − ????2) cos ???? + 2???????? sin ????} .

c) Obtain Fourier series of ????(????) = ????2 in (0, 2????). Hence, deduce that –  (8M)
\frac{\pi^2}{12}=\frac1{1^2}-\frac1{2^2}+\frac1{3^2}-\frac1{4^2}++........

Q. 3) a) Using Bender –Schmidt method, solve \frac{d^2u}{dx^2}-\frac{du}{dt}=0 subject to the conditions, (6M)
????(0, ????) = 0, ????(4, ????) = 0, ????(????, 0) = ????2(16 − ????2) taking ℎ = 1, for 3 minutes.

b) Using convolution theorem, find the inverse Laplace transform of (6M)
F(s)=\frac{S^2+S}{\left(S^2+1\right)\left(S^2+2S+2\right)}

c) Using Residue theorem, evaluate. (5M)
i) \int_0^\pi\frac{d\theta}{2+\cos\theta}
ii) \int_c\frac{z^2}{\left(z+1\right)^2\left(z-2\right)}dz , C: |z| = 1.

Q. 4. a) Solve by Crank –Nicholson simplified formula. (6M)
\frac{d^2u}{dx^2}-16\frac{du}{dt}=0
????(0, ????) = 0, ????(1, ????) = 200????, ????(????, 0) = 0 taking ℎ = 0.25 for one-time step.

b) Obtain the Laurent series which represent the function. (6M)
f\left(Z\right)=\frac{4z+3}{z\left(z-3\right)\left(z+2\right)}
i) 2< |z| <                      ii) |z| > 3

c) Solve (????2 − 3???? + 2)???? = 4ⅇ2???? with ????(0) = −3 and ????′(0) = 5 where D\;\equiv\frac d{dt} (8M)

Q. 5) a) Find the bilinear transformation under which 1, ????, −1 from the z-plane are mapped onto 0,1, ∞ of w-plane.  (6M)

b) Find the Laplace transform of
f(t)\;=\left\{\begin{array}{l}t,\;0\;<\;t\;<\mathrm\pi\\\mathrm\pi-\mathrm t,\;\mathrm\pi<\mathrm t\;<2\mathrm\pi\end{array}\right. and f (t+2π ) = f(t). (6M)

c) Obtain half range Fourier cosine series of ????(????) = ????, 0 < ???? < 2. Using Parseval’s identity, deduce that –(8M)
\frac{\pi^4}{96}=\;\frac1{1^4}+\frac1{3^4}+\frac1{5^4}+......

Q. 6. a) Using contour integration, evaluate: (6M)
\int_{-\infty}^\infty\frac{x^2+x+2}{x^4+10x^2+9}dx

b) Using the least square method, fit a parabola, ???? = ???? + ???????? + ????????2 to the following
data, (6M)

 x -2 -1 0 1 2 y -3.15 -1.39 0.62 2.88 5.378

c) Determine the solution of one-dimensional heat equation \frac{du} {dt}=\;c^2\frac{d^2u}{dx^2} under the boundary conditions ????(0, ????) = 0, ????(????, ????) = 0, ????(????, 0) = ????, (0 < ???? < ????), ???? being the length of the rod.(8M)
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