GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-III (Old) EXAMINATION — WINTER 2019
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Subject Code: 130002
Subject Name: Advanced Engineering Mathematics
Time: 02:30 PM TO 05:30 PM
Date: 22/11/2019
Total Marks: 70
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Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
Q.1
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(a) (i) Solve yexdx + (2y+ex)dy =0
(ii) Solve (x + 1)2 dy/dx —y = 3 (x + 1)2
(b) Obtain Fourier series of f(x) = x2 in the interval(0, 4).
Q.2
(a) (1) Use method of Undetermined coefficients and find general solution of y" + 10y’ + 25y = e-5x
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(b) Find general solution of (D2 + 2D — 35)y = 37 sin 5x
OR
(b) Solve by Variation of parameter method (D2 + 9)y = tan3x
Q.3
(a) Find Fourier series of f(x) =eax in (0,2p), a >0
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(b) Find Fourier series of f(x) = { X , 0 < x < 2; 4-x, 2 < x < 4 }
OR
(a) Find the Series solution of y"' — 2y’ = 0
(b) Express the function f(x) = { sinx, 0 < x < p; 0, p < x } as a Fourier sine integral and show that ?08 sin?xsinp? d? = p/2 sinx
Q.4
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(a) (1) Find Laplace transform of et(1 + vt)2
(ii) Find the inverse Laplace transform of (2s+2) / (s2+2s+10)
(b) State Convolution theorem and using it find inverse Laplace transform of 1/((s—2)(s+2)2)
OR
(a) (i) Find Laplace transform of e-3t u(t — 2)
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(1) Find inverse Laplace transform of e-2s/(s+2)2
(b) Solve initial value problem using Laplace transform method y"' -3y' +2y=12e-x, y(0)=2,y'(0)=6
Q.5
(a) (1) Form Partial differential equation for the equation z=ax+by+ct
(ii) Find Laplace transform of f(t) = { cost , 0 < t < 2p; 0, t > 2p }
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OR
(a) Find the Series solution of 4xy"" + 2y"' +y =0
(b) Using method of Separation of variables solve ?u/?x =4 ?u/?y given that u(0,y) =8e-3y
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