This download link is referred from the post: GTU BE 2020 Summer Question Papers || Gujarat Technological University
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GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- III EXAMINATION - SUMMER 2020
Subject Code: 3130005 Date: 27/10/2020
Subject Name: Complex Variables and Partial Differential Equations
Time: 02:30 PM TO 05:00 PM Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
| Marks | |
|---|---|
| Q.1 (a) If u=x²-3xy is find the corresponding analytic function f(z)=u +iv. | 03 |
| (b) Find the roots of the equation z² —(5+i)z+8+i =0. | 04 |
| (c) (i) Determine and sketch the image of ‘z‘ =1 under the transformation W=z+1. | 03 |
| (ii) Find the real and imaginary parts of f(z) =z̄ +3z. | 04 |
| Q.2 (a) Evaluate ∫(xz -y²)dz) along the parabola y = 2x² from (1,2) to (2,8). | 03 |
| (b) Find the bilinear transformation that maps the points z=∞,i,0 into w=0,i,∞. | 04 |
| (c) (i) Evaluate § ez/(z+1) dz , Where C is the circle |z|=1/2. | 03 |
| (ii) Find the radius of convergence of Σ(1 - 1/n)n² zn. | 04 |
| OR | |
| (c) (i) Find the fourth roots of —1. | 03 |
| (ii) Find the roots of log z= iπ. | 04 |
| Q.3 (a) Find § 1/(z²)dz, where C:|z|=1. | 03 |
| (b) For f(z)= 4/((z-1)²(z-3)), find Residue of f(z) at z=1. | 04 |
| (c) Expand f(z)= 1/((z+2)(z+4)) in a Laurent series for the regions (i) |z| < 2, (ii) 2 < |z| < 4, (iii) |z| > 4. | 07 |
| OR | |
| Q.3 (a) Find § (z²+4)/(z²+2z+5) dz, where C:|z+1|=1. | 03 |
| (b) Evaluate ∫ ez/(z+1)2 dz; C: 4x² +9y² =16 using Cauchy residue theorem. | 04 |
| (c) Expand f(z) = 1/(z(z-1)(z-2)) in Laurent’s series for the regions (i) |z| < 1, (ii) 1 < |z| < 2, (iii) |z| > 2. | 07 |
| Q.4 (a) Derive partial differential equation by eliminating the arbitrary constants a and b from z = ax + by + ab. | 04 |
| (b) (i) Solve the p.d.e. 2r+5s+2t=0. | 03 |
| (ii) Find the complete integral of p² = qz. | 04 |
| OR | |
| (a) Find the solution of x²p + y²q = z². | 03 |
| (b) Form the partial differential equation by eliminating the arbitrary function φ from z = φ(y/x). | 04 |
| (c) (i) Solve the p.d.e. (D² —DD'+D-D')z =0. | 03 |
| (ii) Solve by Charpit’s method yzp² —q =0. | 04 |
| Q.5 (a) Solve (2D² —5DD'+2D'²)z = 24(y — x). | 03 |
| (b) Solve the p.d.e. ut +ux =2(x+ y)u using the method of separation of variables. | 07 |
| (c) Find the solution of the wave equation utt =c²uxx, 0 < x < π with the initial and boundary conditions u(0,t) =u(π,t) =0, t > 0, u(x,0) = k(sin x —sin 2x), ut(x,0) =0, 0 < x < π. (c² =1) | 03 |
| OR | |
| (a) Solve the p.d.e. r+s+t—z=0. | 04 |
| (b) Solve 2ut =uxx +u, given u(x,0)=4e-x using the method of separation of variables. | 07 |
| (c) Find the solution of ut =c²uxx, together with the initial and boundary conditions u(0,t) =u(2,t)=0; t>0 and u(x,0)=10; 0 < x < 2. | 07 |
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This download link is referred from the post: GTU BE 2020 Summer Question Papers || Gujarat Technological University
--- Content provided by FirstRanker.com ---