GTU BE 2019 Summer Question Papers || Gujarat Technological University
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-I & II (NEW) EXAMINATION — SUMMER-2019
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Subject Code: 3110015 Date: 01/06/2019
Subject Name: Mathematics — 2
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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| Marks | |
| Q.1 (a) Find the Fourier integral representation of | 03 |
| F(x) = { x; x ? (0, a) 0; x ? (a, 8) } | |
| (b) Define: Unit step function. Use it to find the Laplace transform of | 04 |
| f(t) = { (t-1)²; t ? (0, 1] 1 ; t ? (1, 8) } | |
| (c) Use the method of undetermined coefficients to solve the differential equation y" - 2y' + y = x²ex. | 07 |
| Q.2 (a) Evaluate ?C F·dr; where F = (x² - y²)i + 2xyj and C is the curve given by the parametric equation C: r(t) = t²i + tj; 0 < t < 2. | 03 |
| (b) Apply Green’s theorem to find the outward flux of a vector field F = (x i + y j) across the curve bounded by y = vx, 2y = 1 and x = 1. | 04 |
| (c) Integrate f(x, y, z) = x - yz² over the curve C = C1 + C2, where C1 is the line segment joining (0, 0, 1) to (1, 1, 0) and C2 is the curve y = x² joining (1, 1, 0) to (2, 4, 0). | 07 |
| OR | |
| (c) Check whether the vector field F = eyz i + xzeyz j + xyeyz k is conservative or not. If yes, find the scalar potential function f(x, y, z) such that F = grad f. | 07 |
| Q.3 (a) Write a necessary and sufficient condition for the differential equation M(x, y)dx + N(x, y)dy = 0 to be exact differential equation. Hence check whether the differential equation [(x + y)ex - ey]dx - xeydy = 0 is exact or not. | 03 |
| (b) Solve the differential equation (x + y²)dx = (ex/y - x)dy | 04 |
| (c) By using Laplace transform solve a system of differential equations x' = 1 - y, y' = x, where x(0) = 1, y(0) = 0. | 07 |
| OR | |
| Q.3 (a) Solve the differential equation (2x³ + 4y)dx - xdy = 0. | 03 |
| (b) Solve (x² + y²)dx + 2xydy = 0 | 04 |
| (c) By using Laplace transform solve a differential equation y'' + 5y' + 6y = e-t, where y(0) = 0, y'(0) = -1. | 07 |
| Q.4 (a) Find the general solution of the differential equation x²y'' - xy' + y = x² | 03 |
| (b) Solve y'' - 7y' - y + 6y = ex | 04 |
| (c) Find a power series solution of the differential equation y'' - xy = 0 near an ordinary point x = 0. | 07 |
| OR | |
| Q.4 (a) Find the general solution of the differential equation xy' + y = 0 | 03 |
| (b) Solve x' = 7x - y, y' = 2x + 5y | 04 |
| (c) Find a Frobenius series solution of the differential equation 2x²y'' + xy' - (x + 1)y = 0 near a regular-singular point x = 0. | 07 |
| Q.5 (a) Write Legendre’s polynomial Pn(x) of degree-n and hence obtain P0(x) and P1(x) in powers of x. | 03 |
| (b) Classify ordinary points, singular points, regular-singular points and irregular-singular points (if exist) of the differential equation y'' + xy = 0. | 04 |
| (c) Solve the differential equation d²y/dx² - 3dy/dx + 2y = x³cosx by using the method of variation of parameters. | 07 |
| OR | |
| Q.5 (a) Write Bessel’s function Jp(x) of the first kind of order-p and hence show that J1/2(x) = v(2/px) sin x. | 03 |
| (b) Classify ordinary points, singular points, regular-singular points and irregular-singular points (if exist) of the differential equation xy'' + y = 0. | 04 |
| (c) Solve the differential equation y'' + 25y = sec 5x by using the method of variation of parameters. | 07 |
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