Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 17 Scheme 17MAT21 Engineering Mathematics II Question Paper
y
?
of
17MAT21
econd Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics -
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing one full question from each module.
Module-1
I a.
P Solve (D
2
+1)y = 3x
2
+ 6x +12. (06 Marks)
-0
6
b. Solve (D
3
+2D
2
+ D)y = e .
a)
(07 Marks)
0
L' c.
By the method of undetermined coefficients, solve (D
2
+ D- 2)y = x +sin x . (07 Marks)
?
,,i ,,
0
.c
,
tD _
0
c.:=
OR
,
_-?A (
'
= 2 a.
Solve (D
2
-6D + 9)y = 6e
3
' +7e
2
'. (06 Marks)
1"
to
II
..v. oc
b.
Solve (D
3
- D)y= (2x +1)+4cosx . (07 Marks)
'-'
2 , I
)
C.
By the method of variation of parameters, solve (D
2
+1)y = cosec x . (07 Marks)
P
cu =
V.
Module-2
0
3 a.
Solve x
2
y"-3xy
s
+ 4y =1+ x
2
. . . (06 Marks)
.:1 0
? -?? ??I
s (73
.
b. Solve xyp
2
-(x
2
+ y
2
)p + xy =0 . (07 Marks)
'
L.. 0
-
a.)
c. Solve (px -y)(py+ x) = a
2
p by taking x = x and y
2
= y. ,
(07 Marks)
71 o
.r:
0 -o
to =
00
OR
.
5
r*
4 a.
P Solve (2+ X)
2
y"+(2+ x)y`+y =sin(2log(2+ x)). (06 Marks)
-6
>
a-, ?,
b. Solve yp
2
+ (x - y)p- x = 0. (07 Marks)
.
E
. 2
<4 ?
C. Obtain the general and singular solution of the equation sin px cosy = cospx sin y+ p .
- co
,.,
D.. c,
E .
(07 Marks)
' rz
.
---
cir
'.- Module-3
3
cr, tz
5 a. Form a partial differential equation by eliminating arbitrary function
o . 6) -6) lx + my +nz =4)(x
2
+ y
2
+ z
2
) (06 Marks)
>,,..
b
?.?
Solve
a2z
.,... .
=x
y
subject to the conditions ?
az
--- log(1+ y) when x - I and z = 0 when x = O.
- ..... .
ax
2
aX
-
,-EJ- 8
> z.. .
0
(07 Marks)
c. Derive an expression for the one dimensional wave equation. (07 Marks)
0 .R'
r
,
i
?? OR
u
6 a. Form a partial differential equation z = f(y + 2x) + g(y-3x) (06 Marks)
a2z
b.
Solve - = z , given that when y = 0, z =ex and -
a,
- - -' (07 Marks)
ay ay
... ,?
c. Find all possible solutions of heat equation u
t
= c
2
u, by the method of separation of
variables. (07 Marks)
)
CC
E.
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.c . Svc left
y
?
of
17MAT21
econd Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics -
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing one full question from each module.
Module-1
I a.
P Solve (D
2
+1)y = 3x
2
+ 6x +12. (06 Marks)
-0
6
b. Solve (D
3
+2D
2
+ D)y = e .
a)
(07 Marks)
0
L' c.
By the method of undetermined coefficients, solve (D
2
+ D- 2)y = x +sin x . (07 Marks)
?
,,i ,,
0
.c
,
tD _
0
c.:=
OR
,
_-?A (
'
= 2 a.
Solve (D
2
-6D + 9)y = 6e
3
' +7e
2
'. (06 Marks)
1"
to
II
..v. oc
b.
Solve (D
3
- D)y= (2x +1)+4cosx . (07 Marks)
'-'
2 , I
)
C.
By the method of variation of parameters, solve (D
2
+1)y = cosec x . (07 Marks)
P
cu =
V.
Module-2
0
3 a.
Solve x
2
y"-3xy
s
+ 4y =1+ x
2
. . . (06 Marks)
.:1 0
? -?? ??I
s (73
.
b. Solve xyp
2
-(x
2
+ y
2
)p + xy =0 . (07 Marks)
'
L.. 0
-
a.)
c. Solve (px -y)(py+ x) = a
2
p by taking x = x and y
2
= y. ,
(07 Marks)
71 o
.r:
0 -o
to =
00
OR
.
5
r*
4 a.
P Solve (2+ X)
2
y"+(2+ x)y`+y =sin(2log(2+ x)). (06 Marks)
-6
>
a-, ?,
b. Solve yp
2
+ (x - y)p- x = 0. (07 Marks)
.
E
. 2
<4 ?
C. Obtain the general and singular solution of the equation sin px cosy = cospx sin y+ p .
- co
,.,
D.. c,
E .
(07 Marks)
' rz
.
---
cir
'.- Module-3
3
cr, tz
5 a. Form a partial differential equation by eliminating arbitrary function
o . 6) -6) lx + my +nz =4)(x
2
+ y
2
+ z
2
) (06 Marks)
>,,..
b
?.?
Solve
a2z
.,... .
=x
y
subject to the conditions ?
az
--- log(1+ y) when x - I and z = 0 when x = O.
- ..... .
ax
2
aX
-
,-EJ- 8
> z.. .
0
(07 Marks)
c. Derive an expression for the one dimensional wave equation. (07 Marks)
0 .R'
r
,
i
?? OR
u
6 a. Form a partial differential equation z = f(y + 2x) + g(y-3x) (06 Marks)
a2z
b.
Solve - = z , given that when y = 0, z =ex and -
a,
- - -' (07 Marks)
ay ay
... ,?
c. Find all possible solutions of heat equation u
t
= c
2
u, by the method of separation of
variables. (07 Marks)
)
CC
E.
17MAT2
Module-4
7 a. Evaluate if r sin 0 dr dO over the cardioids r = a(1 ? cos? above the initial line. (06 Marks)
ii-,
b.
Evaluate
.
1 x dz dx dy (07 Marks)
c. Derive the relation between Beta and Gamma function as B(m, n) =
F(m)F(n)
. (07 Marks)
F(m + n)
OR
8 a.
b.
C.
9 a.
b.
C.
10 a.
b.
Evaluate by changing the order of integration if dydx .
0 Y
Find by double integration, the area lying between the
line y = x.
2
Show that Vcot 0 de =
2
Module-5
(06 Marks)
It
having the period (07 Marks)
transforms. (07 Marks)
r
s+a`
(06 Marks)
s+b,
(07 Marks)
log
(
Find the Laplace transform of t cos 2t
Find the Laplace transform of f(t) = Esin
Solve y" ? 3y' + 2y = 2e" , y(0) = y
1
(0)
Find the inverse Laplace transforms of
By using convolution theorem, find 1.
-1
+
? I e
31
t
cot , 0 < t <
= 0 by using Laplace
OR
s +
+
2s + 2
+ I)(s? l)
(i4)
(
3
4)
(06 Marks)
parabola y = 4x ? x
2
and the
(07 Marks)
(07 Marks
c_ Express f(t) =
sin t,
cos t,
0< t s
7
Y
2
< t TC
< t
in terms of unit step functions and hence find its Laplace
transform_
* * *
Sfx
c
i
o
,
o
,
)?,
cPtIKoDi
?
LiBRAit
y
(07 Marks)
2 oft
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This post was last modified on 01 January 2020