Download OU B-Tech First Year 2012 January 5003 Mathematics II Question Paper

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Time: 3 Hours]
[Max. Marks: 75
Note : Answer all questions from Part A, Answer any five questions from Part B.
PART - A
(25 Marks)

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1. Eliminate arbitrary constant from
y = cx + -
1
c 0 and form a differential equation.
2

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2. Find the solution of the differential equation
3
(y-x+1)dy-(y+x+ 2)dx.- - 0.
3. Show that functions x, x
2

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, x
3
are linearly independent on any interval 1.
2
4. Solve y" + y

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1
- 2y = 0,y(0) = 0, y'(0)=3.
3
5. Find the singular points of x
2

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y" + (x + x
2
)y
1
- y = 0 and classify them. 2

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6. Find the value of T
3
(x) (Chebyshev polynomial).
3
7. Find the value of (9/

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2
, 7/
2
).
2

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8. Express J
3
(x) in terms of J
o
(x) and J

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1
(x). 3
9. Find Laplace transform of 1 + 2
+ 3
2

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S
2
- 35 4
10. Find the inverse Laplace transform of
+

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S
3

3

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PART B (5x10=50 Marks)
11. a) Solve the initial value problem 3x
2
y
4

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dx + 4x
3
y
3
dy = 0, y(1) = 2. 5

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dy
b) Solve the differential equation,
-
d
?

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x
y = y (sin x + cos x).
5
(This paper contains 2 pages)
1

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P.T.O.
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11 1 11111111111 1111 1 1 1 111111111111 1 111111 Code No.: 5003/N
FACULTY OF ENGINEERING AND INFORMATICS
B.E. 1 Year (New) (Common to all Branches) (Suppi.)

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Examination, January 2012
MATHEMATICS ? II
Time: 3 Hours]
[Max. Marks: 75
Note : Answer all questions from Part A, Answer any five questions from Part B.

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PART - A
(25 Marks)
1. Eliminate arbitrary constant from
y = cx + -
1

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c 0 and form a differential equation.
2
2. Find the solution of the differential equation
3
(y-x+1)dy-(y+x+ 2)dx.- - 0.

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3. Show that functions x, x
2
, x
3
are linearly independent on any interval 1.

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2
4. Solve y" + y
1
- 2y = 0,y(0) = 0, y'(0)=3.
3

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5. Find the singular points of x
2
y" + (x + x
2
)y

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1
- y = 0 and classify them. 2
6. Find the value of T
3
(x) (Chebyshev polynomial).

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3
7. Find the value of (9/
2
, 7/
2

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).
2
8. Express J
3
(x) in terms of J

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o
(x) and J
1
(x). 3
9. Find Laplace transform of 1 + 2

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+ 3
2
S
2
- 35 4

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10. Find the inverse Laplace transform of
+

S
3

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3
PART B (5x10=50 Marks)
11. a) Solve the initial value problem 3x
2

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y
4
dx + 4x
3
y

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3
dy = 0, y(1) = 2. 5
dy
b) Solve the differential equation,
-

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d
?
x
y = y (sin x + cos x).
5

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(This paper contains 2 pages)
1
P.T.O.
IMINNE11111111 Code No. : 5003/N
12. a) Find the general solution of the Riccoti equation. 5

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y' = 4xy
2
+ (1- 8x)y + 4x -1, y =1
is a particular solution.
b) Solve the initial value problem y'" - 2y" 5y' + 6y = 0, y(0) = 0, y'(0) = 0, y"(0) =1. 5

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13. a) Solve y" + 4y = cos
2
x.
5
b) If y

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1
= ex is one of the solutions of y" + 3y' - 4y = 0 , then find general solution,
by reducing order of differential equation. 5
14. Find the series solution about x = 0 of the equation (1 - x
2

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) y" - 2xy' + 6y = 0 . 10
0 , min
15. a) Show that Sp
m
(x)Pn(x)dx = 2

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- 1
m = n
2n + 1'
x
b) Evaluate je

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-a
r
-
sin bx dx interms of Gamma function.
16. a) Prove that p (m + 1, n) + p (m, n+ 1) = p (m, n).

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b) Show that J
n
(X) = - lcos (n9 - x sin nO)de
o
5

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5
5
5
17. a) Apply convolution theorem to evaluate L
1

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5

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/,2
"
,2N2

b) Solve (D

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2
+ n
2
) x = a sin (n t+ a ); x = Dx = 0 at t = 0 using . Laplace transform. 5
2

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1,600
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