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:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- 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,?) + iv(r,?). | 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 ? z¯/(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 =ip | 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 f(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(px/l); 0 <x<l | 07 |
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