Download OU B-Tech First Year 2014 June 6003 Mathematics II Question Paper

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Note: Answer all questions from Part-A. Answer any FIVE questions from Part-B.
PART ? A (25 Marks)
1 Form the differential equation by eliminating arbitrary constants a, b from
y = ae
3x

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+ be
5x
,
2 Solve = + x
2

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(IX
3 Solve y" y = 0,, when y = 0 and y = 2 at x = 0.
4 Find the particular integral of (D
2

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+ 1)y = 8e
-x
.
5 Classify the singular points of (I + 2y = 0.
6 prove that P

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n
(l) = 1.
7 Show that J,
;2
(x)

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8 Prove that j" (ix = , ,C' ?1.
0 CI'
(C' +1)
(log (:)(
9 Find the Laplace transform of e

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-t
cost.
+ 2
10 Find inverse Laplace transform of
.s.(s ? 3)(s + 2)

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PART ? B (50 Marks)
11 a) Find the orthogonal trajectories of r = ce
9
, where C is the parameter.
b) Solve

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d y
v = y
2
(sinx + cos x).
dx

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12 a) Using the method of variation of parameters solve (D
2
+1) y = x .
b) Solve (0
2

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? 4D + 2) y = 12e
x
sin2x.
13 Obtain the series solution of the equation
x

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2
y
r,
Ay
r ?

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(x
2
14 a) Prove that 130
.
71,

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?
Fon +
b) Prove that fl
o
(x) .1

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1
(x) dv
1

4)y=0 about x = 0,

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2
------ cos x.
27X
(2)
(3)

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(2)
(3)
(2)
(3)
(2)

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(3)
(2)
(3)
(
5

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)
(5)
(
5
)

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(
5
)
(10)
(

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5
)
(
5
)

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....... . 2
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Code No. 6003 / M
FACULTY OF ENGINEERING and INFORMATICS
B.E. I Year (Main) Examination, June 2014

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Subject : Mathematics ? II
Time : 3 hours Max. Marks : 75
Note: Answer all questions from Part-A. Answer any FIVE questions from Part-B.
PART ? A (25 Marks)
1 Form the differential equation by eliminating arbitrary constants a, b from

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y = ae
3x
+ be
5x
,

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2 Solve = + x
2

(IX
3 Solve y" y = 0,, when y = 0 and y = 2 at x = 0.

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4 Find the particular integral of (D
2
+ 1)y = 8e
-x
.

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5 Classify the singular points of (I + 2y = 0.
6 prove that P
n
(l) = 1.
7 Show that J,

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;2
(x)
8 Prove that j" (ix = , ,C' ?1.
0 CI'
(C' +1)

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(log (:)(
9 Find the Laplace transform of e
-t
cost.
+ 2

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10 Find inverse Laplace transform of
.s.(s ? 3)(s + 2)
PART ? B (50 Marks)
11 a) Find the orthogonal trajectories of r = ce
9

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, where C is the parameter.
b) Solve
d y
v = y
2

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(sinx + cos x).
dx
12 a) Using the method of variation of parameters solve (D
2
+1) y = x .

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b) Solve (0
2
? 4D + 2) y = 12e
x
sin2x.

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13 Obtain the series solution of the equation
x
2
y
r,

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Ay
r ?
(x
2
14 a) Prove that 130

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.
71,
?
Fon +
b) Prove that fl

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o
(x) .1
1
(x) dv
1

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4)y=0 about x = 0,
2
------ cos x.
27X

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(2)
(3)
(2)
(3)
(2)

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(3)
(2)
(3)
(2)
(3)

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(
5
)
(5)
(

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5
)
(
5
)

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(10)
(
5
)
(

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5
)
....... . 2
Code No. 6003 / M
-2?

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15 a) Apply convolution theorem to evaluate
(
5
)
(s

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--
-2 +1)(s2+
b) Use Laplace transform to solve y' y = ex given that y(0) = 1.
(
5

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)
16 a) Find the general solution and singular solution of the Clairaut's equation
(
5
)

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y= (x ? a) p p
2
.
b) Solve the initial value problem ?2y' = 0 with y(0) = 1, ))
1

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(0) = 0.
(
5
)
17 a) Prove that fi),?(x)1)?(x)cly = 0 if m 11 .

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(
5
)
b) Find the Laplace transform of t sin
2

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(3t).
(5)
* * *
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