Download GTU BE/B.Tech 2019 Winter 3rd Sem New 2130002 Advanced Engineering Mathematics Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 3rd Sem New 2130002 Advanced Engineering Mathematics Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III (New) EXAMINATION ? WINTER 2019
Subject Code: 2130002 Date: 22/11/2019

Subject Name: Advanced Engineering Mathematics
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.

MARKS

Q.1 (a) Determine the singular points of the differential equation
22
( 1) '' (2 1) ' 0 x x y x y x y ? ? ? ? ? and classify them as regular or irregular.
03

(b)
(i) Compute
53
,
22
?
?
?
?

(ii) Define (1) Error Function (2) Beta Function
02

02
(c) (i) Solve the I. V. P : '' 4 ' 4 0, (0) 3 & '(0) 1 y y y y y ? ? ? ? ?
(ii) Find (a)
? ?
3
(sin 3 )
t
L e t t ? (b)
1
2
67
5
s
L
s
?
? ?
?
?
?

03

04


Q.2 (a) Solve (1 ) (1 ) 0 x ydx y xdy ? ? ? ? 03
(b)
Solve
2
( 5 6) sin3 D D y x ? ? ?
04

(c)
State the convolution theorem and apply it to evaluate
1
3
1
()
L
s s a
?
?
?
?
?

07
OR
(c) Using Laplace Transformation, Solve '' 6 1, (0) 2, '(0) 0 y y y y ? ? ? ? 07

Q.3 (a)
Find the Laplace transform of
0 , 0 2
()
3 , 2
t
ft
t
? ?
?
?
?
?

03
(b) Find the power series solution of '2 y xy ? . 04
(c)
Obtain Fourier series of the Function
0 , 2 0
()
1 , 0 2
x
fx
x
? ? ? ?
?
?
?
?

07
OR
Q.3
(a)
Find the Inverse Laplace Transform of
2
2
6
( 4)
s
e
s
?
?

03
(b)
Find the series solution of
2
'' 0 y x y ? in power of x .
04
(c) Obtain Fourier series of the Function ( ) | | , f x x x x ? ? ? ? ? ? 07

Q.4 (a)
Solve tan sin 2
dy
y x x
dx
?
03
(b)
Find a sine series for ()
x
f x e ? in 0 x ? ? .
04

(c)
By the Method of Separation of variables , solve 2
uu
u
xt
?
?
?
where
3
( ,0) 4
x
u x e
?
?
07
OR

Q.4 (a)
Solve (2 ) 0, (0) 1
xx
y e dx y e dy y ? ? ? ? ?
03
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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ? III (New) EXAMINATION ? WINTER 2019
Subject Code: 2130002 Date: 22/11/2019

Subject Name: Advanced Engineering Mathematics
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.

MARKS

Q.1 (a) Determine the singular points of the differential equation
22
( 1) '' (2 1) ' 0 x x y x y x y ? ? ? ? ? and classify them as regular or irregular.
03

(b)
(i) Compute
53
,
22
?
?
?
?

(ii) Define (1) Error Function (2) Beta Function
02

02
(c) (i) Solve the I. V. P : '' 4 ' 4 0, (0) 3 & '(0) 1 y y y y y ? ? ? ? ?
(ii) Find (a)
? ?
3
(sin 3 )
t
L e t t ? (b)
1
2
67
5
s
L
s
?
? ?
?
?
?

03

04


Q.2 (a) Solve (1 ) (1 ) 0 x ydx y xdy ? ? ? ? 03
(b)
Solve
2
( 5 6) sin3 D D y x ? ? ?
04

(c)
State the convolution theorem and apply it to evaluate
1
3
1
()
L
s s a
?
?
?
?
?

07
OR
(c) Using Laplace Transformation, Solve '' 6 1, (0) 2, '(0) 0 y y y y ? ? ? ? 07

Q.3 (a)
Find the Laplace transform of
0 , 0 2
()
3 , 2
t
ft
t
? ?
?
?
?
?

03
(b) Find the power series solution of '2 y xy ? . 04
(c)
Obtain Fourier series of the Function
0 , 2 0
()
1 , 0 2
x
fx
x
? ? ? ?
?
?
?
?

07
OR
Q.3
(a)
Find the Inverse Laplace Transform of
2
2
6
( 4)
s
e
s
?
?

03
(b)
Find the series solution of
2
'' 0 y x y ? in power of x .
04
(c) Obtain Fourier series of the Function ( ) | | , f x x x x ? ? ? ? ? ? 07

Q.4 (a)
Solve tan sin 2
dy
y x x
dx
?
03
(b)
Find a sine series for ()
x
f x e ? in 0 x ? ? .
04

(c)
By the Method of Separation of variables , solve 2
uu
u
xt
?
?
?
where
3
( ,0) 4
x
u x e
?
?
07
OR

Q.4 (a)
Solve (2 ) 0, (0) 1
xx
y e dx y e dy y ? ? ? ? ?
03
2
(b)
Find a cosine series for
2
() f x x ? in 0 x ? ? .
04
(c) Using Undetermined co-efficient method , solve the differential equation
23
'' ' 6 6 3 6 y y y x x x ? ? ? ? ?
07

Q.5 (a)
Solve
22
z px qy p q ? ? ?
03

(b)
Find (1)
0
cos
t
t
L e u du
?
?
?
?
?
(2)
1
2
22
2 10
s
L
ss
?
? ?
?
?
?

04
(c)
Find the general solution of P. D. E :
2 2 2
( ) ( ) x yz p y zx q z xy ? ? ? ? ?
07

OR


Q.5 (a)
Form a Partial differential equation from
2
( , ) 0 f xy z x y z ? ? ? ?
03

(b)
Find (1)
00
sin
tt
L au du du
?
?
?
?
(2)
1
2
2
48
s
L
ss
?
? ?
?
?
?

04
(c)
Using Method of Variation of parameters, Solve
3
2
2
( 2 1) 3
x
D D y x e ? ? ?
07

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