Download GTU B.Tech 2020 Summer 3rd Sem 130001 Mathematics Iii Question Paper

Download GTU (Gujarat Technological University Ahmedabad) B.Tech/BE (Bachelor of Technology/ Bachelor of Engineering) 2020 Summer 3rd Sem 130001 Mathematics Iii Previous Question Paper

Seat No.: ________
Enrolment No.___________


GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER? III EXAMINATION ? SUMMER 2020
Subject Code: 130001 Date:26/10/2020
Subject Name: MATHEMATICS-III
Time: 02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.


1
1

Q.1
(a) (i) Solve
x
y
y e
x2
03
04
(ii) Solve 3
x 3 2
xy dx 3 2
3
x y y dy 0


2
d y

(b) Find the power series solution of the equation
x y 0 .
2
dx
07



Q.2
(a) (i) Solve
4x
y 5y 6y e
03
e2x

(ii) Using the method of variation of parameter, solve y 4y 4y
04
x

(b) Using the method of undetermined coefficient, solve y
3y 2
2
y x 4x 8
07


OR


(b) Solve the equation by series method (x )
2
2
y x y 9y 0 about x 0 .
07



Q.3
(a) Find the Fourier series of
2
f (x) x x in the interval (
, ) . Hence, deduce that

2
1
1

1
. . . .
07
6
22
32

(b) Find the Fourier series of f (x) ex , a x a.
07


OR

Q.3
(a) Find the Fourier series of f (x) x , x .
07

(b) Find
the
Half
range
Fourier
cosine
series
of

f (x) x sin x , 0 x . f (x 2 ) f (x)
07



Q.4
(a) (1) Find the Laplace transform of the function f t() e t3
sin t
2 .
03
s 7

(2) Find the i
F s
nverse Laplace transform of the function ( )
.
04
2
s 8 s 25

(b) Solve the differential equation using Laplace Transformation method

d 2 y

y sin t , Given that y(0) ,
1 y (
0) 0, t 0.
dt2
07

OR

Q.4
(a) (1) Find the Laplace transform of the function f t() t
2
cos t
03
6s 4

(2) Find the inverse Laplace transform of the function F (s)
04
2
s 4 s 20

(b)
Define Convolution theorem for Laplace transform. Using Convolution

1

theorem to find Laplace inverse of the function F(s)
2
s a 2
2
07


1

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Q.5
(a) (i) Form the partial differential equation of f 2
2
x y , x y z 0.
03
(ii) Solve ( y z) p (x z) q x y .
04

(b)
2
z
z
z

Solve by the method of separation of variables
2
0
2
x
x y
07


OR
Q.5
(a) (i) Solve p2 q2 x y
03
(ii) Solve pyz z xq x y
04

1 , x 1

(b)
Find the Fourier integral of the function f (x)

0 , x 1

sin cosx
sin

Hence, evaluate (i)
d
(ii)
d
07
0
0
*************
2

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