Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2017 Jan 2016 Dec 15MAT21 Engineering Mathematics II Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2017 Jan 2016 Dec 15MAT21 Engineering Mathematics II Question Paper

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USN
15MAT21

Second Semester B.E. Degree Examination, Dec.2016/Jan.2017
Engineering Mathematics - II
Time: 3 hrs. Max. Marks: 80
Note: Answer FIVE full questions, choosing one full question from each module.
, u
Module-1
x
2
) by inverse differential operator method. (06 Marks) il a. Solve (D-2)
2
y ? 8(e
2
'
+
x
+ x
E
b. Solve (D
2
? 4D + 3) y = e
x
cos 2x, by inverse differential operator method. (05 Marks)
A
e
3.
c. c. Solve by the method of variation of parameters y" ? 6y' + 9y = ?) . (05 Marks)
0
0 x
-
,.
a -
8

czt ?
9
OR

o;
1 2 a. Solve (D
2
? 1)y = x sin 3x by inverse differential operator method. (06 Marks)
03
b. Solve (D
3
? 6D
2
+ 11D ? 6)y =
e 2 x
by inverse differential operator method. (05 Marks)
ii
I
c
T
cv
c. Solve (D
2
+ 2D + 4) y = 2x
2
+ 3 e' by the method of undetermined coefficient. (05 Marks)
.`11
1-)
. =
3 a.
Module-2

Zii ?o-
w
t t
1)

Solve x
3
y"' + 3x
2
y" + xy' + 8y = 65 cos(log x).
= "r'
(06 Marks)
0
b. Solve xy p
2
+ p(3x
2
? 2y
2
) ? 6xy ? 0. (05 Marks)
g
: g .,..
c. Solve the equation y
2
(y ? xp) = x
4
p
2
by reducing into Clairaut's form, taking the substitution
rn co
en m
1
To 8
?. 1 m
,

=
0 ^I=1
OD =

x
?1
and y =
Alia 140
1C
x
(05 Marks)
0
g
.
1_.
-0
,
-.r
0
-r Ts
'
D
o
0?

?
? cd
0L,
l
a.
? F3L
Module-3
' e
.3 5 a. Obtain the partial differential equation by eliminating the arbitrary function.
2
z = f(x + at) + g(x ? at).
:
(06 Marks)
a2z
sin b. Solve ? sm x sin y, for which ? -2 sin y, when x = 0 and z = 0, when y is an odd
axay
ay
multiple of 4. (05 Marks)
(3
2?, _ 2
u

= eL ----
2
- by the method of separation of
at
e
ax
variables. (05 Marks)
(06 Marks)
(05 Marks)
OR
4 a. Solve (2x + 3)
2
y" ? (2x + 3) y' - 12y = 6x. (06 Marks)
b. Solve p
2
+ 4x
5
p ? 12x
4
y = 0. (05 Marks)
c. Solve p
3
? 4xy p + 8y
2
= 0. (05 Marks)
OQ
OR
6 a. Obtain the partial differential equation by eliminating the arbitrary function
Cx + my + nz ? 4)(x
2
+ y
2
+ z
2
).
52z
az
b. Solve z, given that , when y 0 , z = e
x
and ? =
ay ay
c. Find the solution of the wave equation
1 of 2
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Q k L 1._ S - --\)
o
0
3

USN
15MAT21

Second Semester B.E. Degree Examination, Dec.2016/Jan.2017
Engineering Mathematics - II
Time: 3 hrs. Max. Marks: 80
Note: Answer FIVE full questions, choosing one full question from each module.
, u
Module-1
x
2
) by inverse differential operator method. (06 Marks) il a. Solve (D-2)
2
y ? 8(e
2
'
+
x
+ x
E
b. Solve (D
2
? 4D + 3) y = e
x
cos 2x, by inverse differential operator method. (05 Marks)
A
e
3.
c. c. Solve by the method of variation of parameters y" ? 6y' + 9y = ?) . (05 Marks)
0
0 x
-
,.
a -
8

czt ?
9
OR

o;
1 2 a. Solve (D
2
? 1)y = x sin 3x by inverse differential operator method. (06 Marks)
03
b. Solve (D
3
? 6D
2
+ 11D ? 6)y =
e 2 x
by inverse differential operator method. (05 Marks)
ii
I
c
T
cv
c. Solve (D
2
+ 2D + 4) y = 2x
2
+ 3 e' by the method of undetermined coefficient. (05 Marks)
.`11
1-)
. =
3 a.
Module-2

Zii ?o-
w
t t
1)

Solve x
3
y"' + 3x
2
y" + xy' + 8y = 65 cos(log x).
= "r'
(06 Marks)
0
b. Solve xy p
2
+ p(3x
2
? 2y
2
) ? 6xy ? 0. (05 Marks)
g
: g .,..
c. Solve the equation y
2
(y ? xp) = x
4
p
2
by reducing into Clairaut's form, taking the substitution
rn co
en m
1
To 8
?. 1 m
,

=
0 ^I=1
OD =

x
?1
and y =
Alia 140
1C
x
(05 Marks)
0
g
.
1_.
-0
,
-.r
0
-r Ts
'
D
o
0?

?
? cd
0L,
l
a.
? F3L
Module-3
' e
.3 5 a. Obtain the partial differential equation by eliminating the arbitrary function.
2
z = f(x + at) + g(x ? at).
:
(06 Marks)
a2z
sin b. Solve ? sm x sin y, for which ? -2 sin y, when x = 0 and z = 0, when y is an odd
axay
ay
multiple of 4. (05 Marks)
(3
2?, _ 2
u

= eL ----
2
- by the method of separation of
at
e
ax
variables. (05 Marks)
(06 Marks)
(05 Marks)
OR
4 a. Solve (2x + 3)
2
y" ? (2x + 3) y' - 12y = 6x. (06 Marks)
b. Solve p
2
+ 4x
5
p ? 12x
4
y = 0. (05 Marks)
c. Solve p
3
? 4xy p + 8y
2
= 0. (05 Marks)
OQ
OR
6 a. Obtain the partial differential equation by eliminating the arbitrary function
Cx + my + nz ? 4)(x
2
+ y
2
+ z
2
).
52z
az
b. Solve z, given that , when y 0 , z = e
x
and ? =
ay ay
c. Find the solution of the wave equation
1 of 2
1
c Module-5
a. Find i) L {e
3t
(2 cos 5 t ? 3 sin 5 0} ii) L (06 Marks)
{ cos at ? cos bt } .
t

a
u , 2
c. Derive one dimensional heat equation
at
= c-
5
u

ax
2
?
15MAT21
(05 Marks)

I z x+z
Module-4
(x + y + z) dy dx dz. (06 Marks)
7
a. Evaluate
b. Evaluate f f xy dy dx by changing the order of integration.
0 .,
4a
4 -, ,
c. Evaluate x
72
(4 ?x)
512
dx by using Beta and Gamma function.
(05 Marks)
(05 Marks)
OR
8 a. Evaluate
Ei:
e
-tx'
" ) dx dy by changing to polar co-ordinates. Hence show that
f e' dx = M -
D
b. Find by double integration, the area lying inside the circ
k
lelz X
r = a(1- cos 0).
c. Obtain the relation between beta and gamma function in the form
0
(m, n)
= r(in)F(n)
F(n -F n)
Vi
(06 Marks)
Ase
d I
l
ik
i
tsle the c
.
ardioid
(05 Marks)
(05 Marks)
b. If a periodic function of period 2a is defined by
{
f(t) ?
t if 0.__.t.lca.
1
then show that L{f(t)}= tan h(1.
2a ? t if a t 2a s 2
c. Solve the equation by Laplace transform method. y"' + 2y" - y'-2y = 0. Given
y(0) = y' (0) = 0 , y"(0) =6.
OR
10 a. Find U
l
f ,
s?3
}.
s
-
?4s+13
b. Find 1:1
s
, , by using Convolution theorem.
1
2
+a
2
f }
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
c. Express f(t) =
,
sin t, 0 < t < 71
sin 2t, 11 5.. t < 27c
sin 3t, t > 27E
in terms of unit step function and hence find its
Laplace transforms. (05 Marks)
2 of 2
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