Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 15 Scheme 15MAT21 Engineering Mathematics II Question Paper
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Societre
US
b
. 46
Second Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics ? II
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
Solve (D
2
?4D + 4)y = e
2
' + cos2x +4 by inverse differential operator method. (06 Marks)
b. Solve ?
d2y
?2?
dy
+ Sy e
2
x sin x by inverse differential operator method. (05 Marks)
dx` dx
c.
Using the method of undetermined coefficients, solve yff ?3y' + 2y = +ex . (05 Marks)
OR
2
a. Solve
cry
2?
dy
+ y = xex sin x by inverse differential operator method. (06 Marks)
dx dx
b
.
Solve (D
3
6D
2
+ I 1D ? 6)y = e
-2
x + by inverse differential operator method.(05 Marks)
c. Solve- y" ? 2y' +y =
c
by method of variation of parameters.
Module-2
y
3
a. Solve (2x-1)2
cr
+(2x ?l)?
dy
-2y =8x' ?2x +3 .
dx
2
dx
(05 Marks)
(06 Marks)
b. Solve xy(
dy
-2
+y')
dY
+ xy = 0 (05 Marks)
dx
)
dx
Solve x
2
(y ?px)? p
2
y by reducing into Clairaut's form and using the substation X = x
2
and
Y = (05 Marks)
OR
4 a.
Solve x
-l
y"? xy'+ 2y = x sin(log x). (06 Marks)
b. Obtain the general solution of the differential equation p
2
+ 4x'p-12x
4
y = 0 . (05 Marks)
c. Obtain the general and singular solution of y = 2px + p
,
y . (05 Marks)
Module-3
5 a. Form the partial differential equation by eliminating the arbitrary function from the relation
Z = y f(x) + x g(y).
(06 Marks)
(z
axa
b.
Solve = x sin y for which = ?2sin y when x 0 and z 0 when y is an odd
y
()
1
multiple of n/2. (05 Marks)
c. Derive one dimensional wave equation = ?
ax2
(05 Marks)
at2
c.
I of 2
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15MAT21
,.."'"C
.
Societre
US
b
. 46
Second Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics ? II
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a.
Solve (D
2
?4D + 4)y = e
2
' + cos2x +4 by inverse differential operator method. (06 Marks)
b. Solve ?
d2y
?2?
dy
+ Sy e
2
x sin x by inverse differential operator method. (05 Marks)
dx` dx
c.
Using the method of undetermined coefficients, solve yff ?3y' + 2y = +ex . (05 Marks)
OR
2
a. Solve
cry
2?
dy
+ y = xex sin x by inverse differential operator method. (06 Marks)
dx dx
b
.
Solve (D
3
6D
2
+ I 1D ? 6)y = e
-2
x + by inverse differential operator method.(05 Marks)
c. Solve- y" ? 2y' +y =
c
by method of variation of parameters.
Module-2
y
3
a. Solve (2x-1)2
cr
+(2x ?l)?
dy
-2y =8x' ?2x +3 .
dx
2
dx
(05 Marks)
(06 Marks)
b. Solve xy(
dy
-2
+y')
dY
+ xy = 0 (05 Marks)
dx
)
dx
Solve x
2
(y ?px)? p
2
y by reducing into Clairaut's form and using the substation X = x
2
and
Y = (05 Marks)
OR
4 a.
Solve x
-l
y"? xy'+ 2y = x sin(log x). (06 Marks)
b. Obtain the general solution of the differential equation p
2
+ 4x'p-12x
4
y = 0 . (05 Marks)
c. Obtain the general and singular solution of y = 2px + p
,
y . (05 Marks)
Module-3
5 a. Form the partial differential equation by eliminating the arbitrary function from the relation
Z = y f(x) + x g(y).
(06 Marks)
(z
axa
b.
Solve = x sin y for which = ?2sin y when x 0 and z 0 when y is an odd
y
()
1
multiple of n/2. (05 Marks)
c. Derive one dimensional wave equation = ?
ax2
(05 Marks)
at2
c.
I of 2
6 a.
15MAT21
OR
Form a partial differential by eliminating the arbitrary function (I) from the relation
4)(x
2
+y
2
+z
2
,z
2 _ 2xy) = 0 .
(06 Marks)
2
z
Solve
a
+ 4z = 0 , given that when x = 0, z = e''' and ?
az
= 2 ' (05 Marks)
ax
2
ax
au a
21
-
1
c. Determine the solution of the heat equation ? = c
2
by the method of separation of
at aK2
variables for the constant K is positive. (05 Marks)
Module-4
24
7 a. Evaluate if (xy + e )dydx (06 Marks)
I
4a 2\ ax
b. Evaluate dydx by changing the order of integration. (05 Marks)
0 x
2
/4a
C .
Obtain the relation between the beta and gamma function in the form
b.
p(m,n)=
km) -1(0
1( ni+n)
-
X,
OR
8 a. Evaluate fJe
4
'
2
dxdy by changing into polar coordinates.
0 0
X X
Evaluate fe`
-
Y"dzdydx
J
0 0
2
x
y
c. Using beta and gamma function, prove that dx x -
o -x
4
t
, -J1+ x
4
4-\12
Module-5
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
+ t sin in t
(06 Marks) 9 a. Find L
b. If f(t)
t
t 0__tsrm 1 Its
, where g -
7
- t + 2n) = f(t), then prove that L{f(t) = tan h
2 m s
-
2
(05 Marks)
c. Find r
i
S
using convolution theorem.
OR
(05 Marks)
s
` ?
a
2 )2
10 a.
Express f(t) =
0 < t < 1
t 1 < t < 2 in term of unit step function and hence find its Laplace
'
2
t > 2
transform. (06 Marks)
b. Find
L-`
s
2
-6s+13
s +5 1
. (05 Marks)
Employ the Laplace transform to solve the differential equation y"(t)+ 4y'(t)+4y(t)= e'
with the initial condition AO) = 0 and y'(0) = 0.
.
..?...
.i....
Society.,
(05 Marks)
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*
ir% in, A Iti I
CHIKOM
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c.
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This post was last modified on 01 January 2020