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Download GTU B.Tech 2020 Summer 4th Sem 3140610 Complex Variables And Partial Differential Equations Question Paper

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 4th Sem 3140610 Complex Variables And Partial Differential Equations Previous Question Paper

This post was last modified on 04 March 2021

This download link is referred from the post: GTU BE 2020 Summer Question Papers || Gujarat Technological University


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GUJARAT TECHNOLOGICAL UNIVERSITY

SEMESTER-1V EXAMINATION - SUMMER 2020

Subject Code: 3140610 Date:02/11/2020

Subject Name: Complex Variables and Partial Differential Equations

Time: 10:30 AM TO 01:00 PM Total Marks: 70

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Instructions:

  1. Attempt all questions.
  2. Make suitable assumptions wherever necessary.
  3. Figures to the right indicate full marks.
Mark
Q.1 (a) Show that the function u=x²-y² is harmonic and find the corresponding analytic function. 03
(b) Find the fourth roots of —1. 04
(c) (1) Find the image of the infinite strip 0 < x < 1 under the transformation w=iz +1. Sketch the region. 03
(i1) Write the function f(z)=z+ 1/z in the form f(z) = u(r,θ) + iv(r,θ). 04
Q.2 (a) Evaluate ∫(xz + ixy)dz from (1,1) to (2,4) along the curve x=t,y =t². 03
(b) Find the bilinear transformation which transforms z=2,1,0 into w=1,0,i 04
(c) (1) Evaluate ∫ z̄/(z-2) dz, where C is |z + i| =1, counter clockwise. 03
(i1) Find the radius of convergence of ∑ zn/(2+i)n. 04
OR
(c) (i) Find the roots of the equation z² + 2iz +(2 —4i) = 0. 03
(i1) Find the roots of log z =iπ 04
Q.3 (a) Find ∫ ez/(z-i)2(z-0) dz, where C:|z| = 1/2. 03
(b) Find the residues of f(z) = 1/((z-1)²(z-3)), has a pole at z = 3 and a pole of order 2 at z=1. 04
(c) Expand f(z) = 1/((z+1)(z+3)) in a Laurent series for the regions (l)|z| <1, (il 1<|z| <3, (iii)|z| > 3. 07
OR
Q.3 (a) Evaluate ∫ dz/(z+i), where C: |z - 1| =1. 03
(b) Evaluate by using Cauchy’s residue theorem ∫ dz/(z(z-2)). 04
(c) Expand f(z) = 1/(z(z-2)) in Laurent’s series for the regions (i)|z| <1, (ii) 1<|z| <2, (iii)|z| > 2. 07
Q.4 (a) Solve yq-xp=z. 03
(b) Derive partial differential equation by eliminating the arbitrary constants a and b from z = (x² +a)(y² +b). 04
(c) (i) Solve the p.d.e. r—3rs +2t =0. 03
(i1) Find the complete integral of p(1+q) = qz. 04
OR
Q.4 (a) Find the solution of (y—z)p+(z—x)q=x—y. 03
(b) Form the partial differential equation by eliminating the arbitrary function from φ(x+y+z,x² +y² +z²)=0. 04
(c) (i) Solve the p.d.e. s+p—q=z+xy. 03
(ii) Solve by Charpit’s method q =3p². 04
Q.5 (a) Solve (D² +2DD'+D'²)z = ex+y 03
(b) Solve the p.d.e. ut =4uxx ,u(0,y) = 8e-2y using the method of separation of variables. 04
(c) Find the solution of the wave equation utt =c²uxx, 0<x<L with the conditions u(0,t) = u(L,t) = 0;t > 0, u(x,0)= x, ut(x,0)=0:0<x<L. 07
OR
Q.5 (a) Solve the p.d.e. r+s+q—z=0. 03
(b) Solve xux —2yuy =0 using the method of separation of variables. 04
(c) Find the solution of ut =c²uxx, together with the initial and boundary conditions u(0,t) = u(l,t) = 0;t ≥ 0 and u(x,0) = sin(πx/l); 0 <x<l 07

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This download link is referred from the post: GTU BE 2020 Summer Question Papers || Gujarat Technological University

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