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GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER-III (OLD) EXAMINATION - WINTER 2018Subject Code:130002
Subject Name:Advanced Engineering Mathematics
Date:17/11/2018
Time:10:30 AM TO 01:30 PM
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Total Marks: 70Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1 (a) (1) Solve: (x tan (y/x) - ysec2(y/x))dx + xsec2(y/x)dy = 0 04
(ii) Find Laplace transform of (cos at - cos bt)2 03
(b) (i) Find Inverse Laplace transform of 3(s2 -1)2 04
(ii) Find Half-Range cosine series for f(x) = 1; 0 < x < 1 03
= x; 1 < x < 2.
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OR
Q.1 (a) Solve: (y + z)p + (x+z)q = x+y 07
(b) Obtain Fourier series for f(x) = x-x2; -1 < x < 1 07
Q.2 (a) Find series solution of d2y/dx2 + x2(dy/dx) - y = 0. 04
(b) (i) Solve 2r + 5s + 2t = 0. 03
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(ii) Solve: (2xy+y2-tany)dx+(x2 - xtan2y +sec2y)dy =0. 07
OR
Q.2 (a) Obtain Fourier series for f (x) = |Sinx|; -p < x < p 04
(b) (i) Define periodic function. Find Laplace transform of f(t)=1; 0<t<2, f(t+2) = f(t). 03
Q.3 (a) (i) Solve: (x2-1)(dy/dx)+2xy =1. 04
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(ii) State Convolution theorem. Use it find Inverse Laplace transform of 1/((s2 +a2)(s2 +b2)) 03(b) (i) Solve: d2x/dt2 +4x=0. 04
(ii) Evaluate: ?ab (e-at - e-bt)/t dt Prove that I = ln(b/a) 03
OR
Q.3 (a) (i) Solve ?u/?t = c2 ?2u/?x2 using method of separable variable method. 04
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(ii) Solve p2 + q2 =1 03Q.4 (a) (i) Solve: (D3 -6D2 +11D-6)y=0. 04
(ii) Find Inverse Laplace transform of (5s2 -2s-19)/((s+3)(s-1)2) 03
OR
Q.4 (b) Find series solution of (1 +x2)(d2y/dx2) +x(dy/dx) -y=0. 07
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Q.5 (a) (i) Express the function f(x) = 1; |x| < 1 as Fourier integral. 05
= 0; Otherwise
(ii) Define Dirac-Delta function 02
(b) Find solution of d2y/dx2 +2y=x2tanx using method of variation of parameter. 07
OR
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Q.5 (a) (i) Solve: (x2D2 +4xD + 2)y =x+logx. 05
(ii) Define Gamma function and find its value for 3/2. 02
(b) Find solution of (D2 + D)y = x2 +2x +4 using method of undetermined coefficient 07
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