Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) 3110015 Mathematics ?2 Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 1st And 2nd Sem (New And SPFU) 3110015 Mathematics ?2 Previous Question Paper

1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 3110015 Date: 01/06/2019

Subject Name: Mathematics ?2

Time: 10:30 AM TO 01:30 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) Find the Fourier integral representation of
?? (?? ) = {
?? ; ?? ? (0, ?? )
0 ; ?? ? (?? , ?)

03
(b) Define: Unit step function. Use it to find the Laplace transform of
?? (?? ) = {
(?? ? 1)
2
; ?? ? (0, 1]
1 ; ?? ? (1, ?)

04
(c) Use the method of undetermined coefficients to solve the differential
equation ?? ??
? 2?? ?
+ ?? = ?? 2
?? ?? .
07

Q.2 (a)
Evaluate ? ?? ?
? ?? ?? ? ;
?? where ?? ?
= (?? 2
? ?? 2
)?? ? + 2???? ?? ? and C is the curve
given by the parametric equation
?? ? ?? (?? ) = ?? 2
?? ? + ?? ?? ? ; 0 ? ?? ? 2 .
03
(b) Apply Green?s theorem to find the outward flux of a vector field ?? ?
=
1
????
(?? ?? ? + ?? ?? ? ) across the curve bounded by ?? = ? ?? , 2?? = 1 and ?? = 1.
04
(c) Integrate ?? (?? , ?? , ?? ) = ?? ? ?? ?? 2
over the curve ?? = ?? 1
+ ?? 2
, where C1 is
the line segment joining (0,0,1) to (1,1,0) and C2 is the curve y=x
2
joining
(1,1,0) to (2,4,0).
07
OR
(c)
Check whether the vector field ?? ?
= ?? ?? +2?? ?? ? + ?? ?? ?? +2?? ?? ? + 2?? ?? ?? +2?? ?? ?
is
conservative or not. If yes, find the scalar potential function ?? (?? , ?? , ?? ) such
that ?? ?
= grad ?? .
07

Q.3 (a) Write a necessary and sufficient condition for the differential equation
?? (?? , ?? )???? + ?? (?? , ?? )???? = 0 to be exact differential equation. Hence check
whether the differential equation
[(?? + 1)?? ?? ? ?? ?? ]???? ? ?? ?? ?? ???? = 0
is exact or not.
03
(b) Solve the differential equation
(1 + ?? 2
)???? = (?? ? tan
?1
?? ? ?? )????
04
(c)
By using Laplace transform solve a system of differential equations
????
????
=
1 ? ?? ,
????
????
= ??? , where ?? (0) = 1, ?? (0) = 0.
07
OR
Q.3 (a) Solve the differential equation
(2?? 3
+ 4?? )???? ? ?????? = 0.
03
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1
Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER ?I &II (NEW) EXAMINATION ? SUMMER-2019
Subject Code: 3110015 Date: 01/06/2019

Subject Name: Mathematics ?2

Time: 10:30 AM TO 01:30 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) Find the Fourier integral representation of
?? (?? ) = {
?? ; ?? ? (0, ?? )
0 ; ?? ? (?? , ?)

03
(b) Define: Unit step function. Use it to find the Laplace transform of
?? (?? ) = {
(?? ? 1)
2
; ?? ? (0, 1]
1 ; ?? ? (1, ?)

04
(c) Use the method of undetermined coefficients to solve the differential
equation ?? ??
? 2?? ?
+ ?? = ?? 2
?? ?? .
07

Q.2 (a)
Evaluate ? ?? ?
? ?? ?? ? ;
?? where ?? ?
= (?? 2
? ?? 2
)?? ? + 2???? ?? ? and C is the curve
given by the parametric equation
?? ? ?? (?? ) = ?? 2
?? ? + ?? ?? ? ; 0 ? ?? ? 2 .
03
(b) Apply Green?s theorem to find the outward flux of a vector field ?? ?
=
1
????
(?? ?? ? + ?? ?? ? ) across the curve bounded by ?? = ? ?? , 2?? = 1 and ?? = 1.
04
(c) Integrate ?? (?? , ?? , ?? ) = ?? ? ?? ?? 2
over the curve ?? = ?? 1
+ ?? 2
, where C1 is
the line segment joining (0,0,1) to (1,1,0) and C2 is the curve y=x
2
joining
(1,1,0) to (2,4,0).
07
OR
(c)
Check whether the vector field ?? ?
= ?? ?? +2?? ?? ? + ?? ?? ?? +2?? ?? ? + 2?? ?? ?? +2?? ?? ?
is
conservative or not. If yes, find the scalar potential function ?? (?? , ?? , ?? ) such
that ?? ?
= grad ?? .
07

Q.3 (a) Write a necessary and sufficient condition for the differential equation
?? (?? , ?? )???? + ?? (?? , ?? )???? = 0 to be exact differential equation. Hence check
whether the differential equation
[(?? + 1)?? ?? ? ?? ?? ]???? ? ?? ?? ?? ???? = 0
is exact or not.
03
(b) Solve the differential equation
(1 + ?? 2
)???? = (?? ? tan
?1
?? ? ?? )????
04
(c)
By using Laplace transform solve a system of differential equations
????
????
=
1 ? ?? ,
????
????
= ??? , where ?? (0) = 1, ?? (0) = 0.
07
OR
Q.3 (a) Solve the differential equation
(2?? 3
+ 4?? )???? ? ?????? = 0.
03
2
(b)
Solve: (?? + 1)
????
????
? ?? = ?? 3?? (?? + 1)
2
.
04
(c)
By using Laplace transform solve a differential equation
?? 2
?? ?? ?? 2
+ 5
????
????
+ 6?? =
?? ??? , where ?? (0) = 0, ?? ?
(0) = ?1.
07

Q.4 (a) Find the general solution of the differential equation
?? ??? ????
????
+
?? ??? ?? =
1
?? 2

03
(b)
Solve :
?? 3
?? ?? ?? 3
? 7
????
????
+ 6?? = ?? ??
04
(c) Find a power series solution of the differential equation ?? ??
? ???? = 0 near
an ordinary point x=0.
07
OR
Q.4 (a) Find the general solution of the differential equation

????
????
+
?? ?? ? ??? = 0.
03
(b)
Solve : ?? 3
?? 3
?? ?? ?? 3
+ 2?? 2
?? 2
?? ?? ?? 2
+ 2?? = ??
04
(c) Find a Frobenius series solution of the differential equation 2?? 2
?? ??
+ ?? ?? ?
?
(?? + 1)?? = 0 near a regular-singular point x=0.
07

Q.5 (a) Write Legendre?s polynomial ?? ?? (?? ) of degree-n and hence obtain ?? 1
(?? )
and ?? 2
(?? ) in powers of x.
03
(b) Classify ordinary points, singular points, regular-singular points and
irregular-singular points (if exist) of the differential equation ?? ??
+ ?? ?? ?
=
0.
04
(c) Solve the differential equation
?? 2
?? 2
?? ?? ?? 2
? 2?? ????
????
+ 2?? = ?? 3
cos ??
by using the method of variation of parameters.
07
OR
Q.5 (a) Write Bessel?s function ?? ?? (?? ) of the first kind of order-p and hence show
that ?? 1/2
(?? ) = ?
2
????
sin?? .
03
(b) Classify ordinary points, singular points, regular-singular points and
irregular-singular points (if exist) of the differential equation ????
??
+ ?? ?
=
0.
04
(c) Solve the differential equation ?? ??
+ 25?? = sec 5??
by using the method of variation of parameters.
07


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