Enrolment No./Seat No_______________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE- SEMESTER–I & II EXAMINATION – WINTER 2024
Subject Code:3110015
Subject Name:Mathematics - 2
Time:10:30 AM TO 01:30 PM
Date:10-01-2025
Total Marks:70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
4. Simple and non-programmable scientific calculators are allowed.
Q.1
(a) Find a such that  x  3 y  iˆ   y  2z  ˆj   x  az  kˆ is solenoidal.
(b) Solve ye x dx   2 y  e x  dy  0
03
(c) Verify Green’s theorem in a plane for the integral  x  2 y  dx  x dy  taken


07
04
C
around the circle x  y  4.
2
Q.2
2
(a) Find the Laplace transform of  sin 2t  cos 2t 2 .
(b) Find the Fourier sine integral of f ( x)  ebx .
03
(c) Using the Frobenius method, obtain the series solution for
07
2 x 1  x  y  1  x  y  3 y  0 about x0  0 .
04
OR
Q.3
(c) Using the method of undetermined coefficients, solve
 D2  9 y  x  e2 x  sin 2x .
07
(a) Find the arc length of the curve r (t )  t 2iˆ  t 3 ˆj between (1,1) and (4,8).
(b)
et sin t
Find the Laplace transform of
.
03
(c) State the convolution theorem and verify it for f (t )  t and g (t )  e2t .
07
04
t
Q.3
Q.4
OR
(a) Find the inverse Laplace transform of tan 1 s .
(b) Solve x2 p2  3xyp  2 y 2  0 .
(c) Solve the initial value problem using Laplace transformation y  3 y  2 y  4t
with y(0)  1, y(0)  1 .
(a)
Find the inverse Laplace transform of
(b) Solve  D2  1 y  e x .
Q.4
e  s
.
s 2  2s  2
03
04
07
03
04
(c) Using method of variation of parameters, solve  D2  2 D  2  y  e x tan x.
07
OR
(a) Classify the singular points of the equation x3  x  2  y  x3 y  6 y  0 .
03
1
(b)
Q.5
Find the Laplace transform of sin t .
(c) Find the series solution of 1  x 2  y  xy  9 y  0 .
04
(a) Solve dr   2r cot   sin 2  d  0
sin x
(b) If
is one of the solution of
03
y1 
(c)
x
Show that (i) J 1 ( x) 
2
Q.5
(a)
xy  2 y  xy  0 , find the second solution.
2
2
sin x (ii) J 1 ( x) 
cos x .

x
x
2
OR
t t
Find the Laplace transform of   sin at dt dt .
(b) Solve  x2 D2  xD  2  y  6 .
07
04
07
03
0 o
04
(c) Prove that J 0 2 ( x)  2  J12 ( x)  J 2 2 ( x)  J 32 ( x)  ...  1


07
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2