GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-III (NEW) EXAMINATION — SUMMER 2019
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Subject Code: 2130002 Date: 30/05/2019Subject Name: Advanced Engineering Mathematics
Time: 02:30 PM TO 05:30 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) Solve (x+y—2)dx+(x—y+4)dy=0 03
(b) Solve (1+y2)dx + (x—etan-1y)dy=0 04
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(c) Expand f(x)= |cos x| as a Fourier series in the interval -p < x < p 07
Q.2 (a) Define unit step function and unit impulse function. Also sketch the graphs. 03
(b) Solve (D2 + 2D + 1)y = 4sin2x 04
(c) Find the series solution of y” + xy’ + y =0 about the ordinary point x=0. 07
OR
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(c) Find the Fourier series expansion for f(x), if
f(x) = { -p, -p < x < 0 ; x, 0 < x < p } Also deduce that 1/12 + 1/32 + 1/52 + ... = p2/8 07
Q.3 (a) Using Fourier integral representation, show that
?08 cos?x sin?x d? = { p/2, 0 < x < 8 ; 0, x > 8 } 03
(b) solve (D2 + y) = x2sin2x 04
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(c) Solve by method of variation of parameters (D2 + 9y) = 1/(1+sin3x) 07
OR
Q.3 (a) Find Laplace transform of xeat sin at 03
(b) Solve (D2 - D)y = 5ex - sin2x 04
(c) Solve x2y” - xy’ + 4y = cos(log x) + xsin(log x) 07
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Q.4 (a) Find the orthogonal trajectories of the curve Y = x2 + c 03
(b) Find the Laplace transform of (i) cos(at + b) (ii) sin2 3t 04
(c) State convolution theorem and apply it to evaluate L-1 { s2 / (s2 + 4)2 } 07
OR
Q.4 (a) Solve (D3 - 3D2 + 4)y = 0 03
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(b) Find Half range cosine series for f(x)= (x—1)2 in the interval 0 < x < l 04
(c) Solve y”+4y’+3y=e-t, y(0)=y'(0) =1 using Laplace transform. 07
Q.5 (a) Form the partial differential equation by eliminating the arbitrary constants from z = ax + by + a2+b2 03
(b) Solve (y—z)p+(x—y)q=z—x 04
(c) Solve ?u/?t = 3?2u/?x2, where u(x,0)=4e-x using the method of separation of variables. 07
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OR
Q.5 (a) Form the partial differential equation by eliminating the arbitrary function from f(x2 +y2, z—xy)=0 03
(b) Solve (?2z/?x?y) = x+ y. 04
(c) A bar with insulated sides is initially at temperature 0°C throughout. The end x = 0 is kept at 0° C and heat is suddenly applied at the end x = l so that ?u/?x = A for x = l, where A is a constant. Find the temperature function. 07
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