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Download GTU BE/B.Tech 2019 Winter 3rd Sem New 3130005 Complex Variables And Partial Differential Equations Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 3rd Sem New 3130005 Complex Variables And Partial Differential Equations Previous Question Paper

This post was last modified on 20 February 2020

GTU BE/B.Tech 2019 Winter Question Papers || Gujarat Technological University


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Seat No.:

Subject Code: 3130005

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER- III (New) EXAMINATION — WINTER 2019

Subject Name: Complex Variables and Partial Differential Equations

Time: 02:30 PM TO 05:00 PM

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Instructions:

  1. Attempt all questions.
  2. Make suitable assumptions wherever necessary.
  3. Figures to the right indicate full marks.

Q1

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(a) State De-Moivre’s formula and hence evaluate (3i/(2+3i))100.

(b) Define harmonic function. Show that u(x,y) = sinhx siny is harmonic function, find its harmonic conjugate v(x,y).

(c) Find the real and imaginary parts of f(z) = (1+iv3)n + (1-iv3)n.

Q2

(a) Determine the Mobius transformation which maps z1 = 0, z2 = 1, z3 = 8 into w1 = —1, w2 = —i, w3 = 1.

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(b) Define logz , prove that ii = e-(p/2 + 2np).

(c) Expand f(z) = 1/(z-1)(z-2) valid for the region (i) |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2

OR

(c) Find the image of the infinite strips (i) 0 < y < 1/4 (ii) 1/4 < y < 1/2 under the transformation w = 1/z. Show the region graphically.

Q3

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(a) Evaluate ?c(x —y +ix2)dz where c is a straight line from z =0 to z = 1+i.

(b) Check whether the following functions are analytic or not at any point, (i) f(z)=3x+y+i(3y—x) (ii) f(z)=zn.

(c) Using residue theorem, evaluate ?08 dx/(x2+1)2.

OR

(a) Expand Laurent series of f(z) = (1-e2z)/z4 at z=0 and identify the singularity.

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(b) If f(z) = u + iv, is an analytic function , prove that ((?2/?x2) + (?2/?y2)) |Ref(z)|2 = 2|(f'(z))|2.

(c) Evaluate the following:

  1. ?c z2 dz where c is the circle (a) |z| = 2 (b) |z| = 3.
  2. ?c Sinz/z2 dz where c is the circle |z| = 1.

Date: 26/11/2019

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Total Marks: 70

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Q.4

(a) Evaluate ?01 Re(z)dz along the curve z(t) = t + it2.

(b) Solve x2p + y2q = (x+y)z

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(c) Solve the equation ?u/?t = k ?2u/?x2 for the condition of heat along rod without radiation subject to the conditions (i) u = 0 for x = 0 and x = l; (ii) u = lx—x2 at t = 0 for all x.

OR

(a) Solve ?2z/?x2 + 2?2z/?x?y + ?2z/?y2 = e2x+3y.

(b) Solve px + qy = pq using Charpit’s method.

(c) Find the general solution of partial differential equation uxy = 9uxx using method of separation of variables.

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Q.5

(a) Using method of separation of variables, solve ?u/?x = 2 ?u/?t + u

(b) Solve z(xp —yq) = y2 —x2.

(c) A string of length L = p has its ends fixed at x = 0 and x = p. At time t = 0, the string is given a shape defined by f(x) = 50x(p — x) , then it is released. Find the deflection of the string at any time t.

OR

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(a) Solve p3+ q3 =x+y.

(b) Find the temperature in the thin metal rod of length l with both the ends insulated and initial temperature is sin px / l

(c) Derive the one dimensional wave equation that governs small vibration of an elastic string . Also state physical assumptions that you make for the system.

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