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

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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+i√3)ⁿ + (1-i√3)ⁿ.

Q2

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

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(b) Define logz , prove that iⁱ = e^{-(π/2 + 2nπ)}.

(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 ∫₀^∞ dx/(x²+1)².

OR

(a) Expand Laurent series of f(z) = (1-e^2z)/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 = e^2x+3y.

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

(c) Find the general solution of partial differential equation u_xy = 9u_xx 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 = π has its ends fixed at x = 0 and x = π. At time t = 0, the string is given a shape defined by f(x) = 50x(π — 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 πx / 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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