B.E. APPLIED MATHEMATICS- III-Mumbai-December 2017

MUMBAI UNIVERSITY

Subject: APPLIED MATHEMATICS- III

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 f(t) = te^-3t sint. (5M)

b) Obtain Complex form of Fourier series of f (x)= e^x , -1< x< 1 in (-1, 1).  (5M)

c) Does there exist an analytic function whose real part is u = k (1+ cos Θ )? Give justification. (5M)

d) The equations of lines of regression are 3x + 2y = 26 and 6x + y = 31. Find i) means of x and y, ii) coefficient of correlation between x and y. (5M)

Q. 2) a) Evaluate \int_0^\infty e^t\sin\;2t\;\cos\;\;3t\;dt. (6M)

b Find the image of the square bounded by lines x = 0,x=2, y = 0, y = 2 in the z-plane under the transformation w= (1 + I) z +2 -i. (6M)

c) Obtain Fourier series of f(x) = |x|  in (-π, π). Hence, deduce that –
\frac{\pi^2}8=\frac1{1^2}+\frac1{3^2}+\frac1{5^2}+........

Q. 3 a) Find the inverse Laplace transform of F(s)=\frac s{\left(s^2+9\right)\left(s^2+4\right)}  (6M)

b) Solve \frac{d^2u}{dx^2}-100\;\frac{du}{dt}=0  with u(0, t) =0 u(1, t) = 0, u(x, 0) = x (1 – x) taking h = 0.1 for three time steps up to t = 1.5 by Bender-Schmidt method. (6M)

c) Using Residue theorem, evaluate  (8M)

i) \int_0^{2\mathrm\pi}\frac{d\theta}{5+4\;\cos\theta}

ii) \int_{-\infty}^\infty\frac{dx}{\left(x^2+1\right)^2}

Q. 4) a) Solve hy Crank Nicholson simplified formula \frac{d^2u}{dx^2}-\frac{du}{dt}=0 u(0.1) = 0, u(5,t) = 100, u(x, 0) = 20 taking h= 1 for one time step. (6M)

b) Obtain the Taylor’s and Laurent series which represent the function. (6M)

f(z)=\frac z{\left(z-1\right))z-2)}  in the region, I |z| < 1 ii 1< |z| , 2

c) Solve (D² – 3D + 2)y = 4e^2t with y) = -3.y’ (0) = 5 where D\equiv\;\frac d{dt} (8M)

Q. 5) a) Find an analytic function f(x) = u + iv, if u= e^-x{(x²-y²) cos y + 2 xy sin y } . (6M)

b) Find the Laplace transform of f(t)=t\;\sqrt{1+\;\sin\;t} (6M)

c) Obtain half range Fourier cosine series of f(x)=x 0< x < 2. Using Parseval’s identity, deduce that –

\frac{\pi^4}{96}=\frac1{1^4}+\frac1{3^4}+\frac1{5\left(4\right)}+....... (8M)

Q. 6) a) If f(a)=\oint\frac{3z^2\;+\;7z\;+\;1}{z-a}dz (6M)

 C : x² + y² = 4 find the values of f (3).f’ (1-i) and f”(1-i)

b) Find the coefficient of correlation between height of father and height of son from the following data, (6M)

c) A lightly stretched string with fixed endpoints x = 0 and x = 1, in the shape defined by y = kx (1-x) where k is a constant is released from this position of test Find y(x, t), the vertical displacement if \frac{d^2y}{dt^2}=\;C^{2\;}\frac{d^2y}{dx2}   (8M)
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