Code No.

: BS-259
Roll No. ......................... Total No. of Printed Pages : 10 1.

Code No. : BS-259 Write statement of first translation theorem.

2.
Online Annual Examination, 2022
Write definition of Inverse Laplace Transform.
B.Sc. Part II
MATHEMATICS 3. z = px + qy + pq
Paper II
[Differential Equation] Write complete integral of differential equation
Time : Three Hours ] [ Maximum Marks : 50 z = px + qy + pq.

4.

Write definition of Lagrange’s Linear equation.

5. r + a2t = 0
Note : Section ‘A’, containing 10 very short answer type
Write order of partial differential equation r + a2t
questions, is compulsory. Section ‘B’ consists of
short answer type questions and Section ‘C’ = 0.
consists of long answer type questions. Section ‘A’ 6.
has to be solved first.

Write definition of partial differential equation of

Section ‘A’ second order.

7.

Answer the following very short answer type Write definition of power series.
questions in one or two sentences. 1 × 10 = 10
P. T. O.
 Code No. : BS-259 Code No. : BS-259
8. n- Solve :

Write Bessel’s differential equation of order-n.  p 1 

L–1  2 
 p  6 p  25 
9.
2.
Write definition of continuity of a functional.
xzp + yzq = xy
10.
Solve :
What do you mean by proper field.
xzp + yzq = xy

Section ‘B’
Or
150-200

Answer the following short answer type questions z(p2 – q2) = x – y

with word limit 150-200. 3 × 5 = 15
Find complete integral :
1. L{e+ sin2 t}
z(p2 – q2) = x – y
Solve : L{e+ sin2 t}
3.

(D2 – 2DD + D2)z = 12xy

Or
Solve :

 p 1  (D2 – 2DD + D2)z = 12xy

L–1  2 
 p  6 p  25 

P. T. O.
 Code No. : BS-259 Code No. : BS-259
I[1] = 1, I[x] = 2
Or
5 1
I[x2] =  sinh 1 2
2 4
(2D2 – 5DD + 2D2)z = 24(y – x) Let a functional I[y(x)] defined on the class C[0, 1]
Solve : be given by
(2D2 – 5DD + 2D2)z = 24(y – x) 1

4.
I[y(x)] = 0 1  [ y( x)]2 dx

d2y Prove that : I[1] = 1, I[x] = 2 and

 y 0
dx 2
5 1
Solve by power series method : I[x2] =  sinh 1 2
2 4
d2y
 y 0
dx 2
Or

Or
1
I [ y ( x)]   ( y2  y  1)dx,
0
2 J n ( x)  J n1 ( x)  J n1 ( x)
y(0) = 1, y(1) = 2
Prove that :
2 J n ( x)  J n 1 ( x)  J n 1 ( x) Test the extremal of the following functional :

5. I[y(x)] C[0, 1] 1
I [ y ( x)]   ( y2  y  1)dx,
0

1 y(0) = 1, y(1) = 2
I[y(x)] = 0 1  [ y( x)]2 dx
P. T. O.
 Code No. : BS-259 Code No. : BS-259

Section ‘C’ Or

300-350

p+q=x+y+z
Answer the following long answer type questions
Solve :
with word limit 300-350. 5 × 5 = 25
p+q=x+y+z

1. (D2 + 9) y = cos 2t, y(0) = 1, y    1.
2 3.

 y2 x2
Solve : (D2 + 9) y = cos 2t, If y(0) = 1, y    1. y 2 r  2 xys  x 2t  p q 0
2 x y

Classify the following equation and solve it :

Or
y2 x2
y 2 r  2 xys  x 2t  p q 0
 1  x y
0 cos x 2 dx = 2 2
 1 
Show that 0 cos x 2 dx = 2 2
· Or

2.

(p2 + q2)y = qz r = a2t

Solve by charpit’s method : Solve by monge’s method :

(p2 + q2)y = qz r = a2t

P. T. O.
 Code No. : BS-259 Code No. : BS-259
4. Find the extremals of the functional
/ 2
pn1 ( x)  pn1 ( x)  (2n  1) pn ( x) I[y, z] = 0 ( y2  z2  2 yz )dt
xpn  pn1  (n  1) pn Subject to the boundary conditions
Prove that :  
y(0) = 1, y    0, z(0) = – 1 and z    0 .
(a) pn1 ( x)  pn1 ( x)  (2n  1) pn ( x) 2 2

(b) xpn  pn1  (n  1) pn

Or

b
Or I[y(x)] = a ( x 2 y2  2 y 2  2 xy)dx
d2y
 y  0, y(0)  0, y ( z )  0
dx 2 Find the extremal of the functional :

b
Obtain the eigen values and eigen functions of the I[y(x)] = a ( x 2 y2  2 y 2  2 xy)dx
d2y
sturm Liouville problem  y  0, y(0) = 0, y(z) d
dx 2
= 0.

/ 2
5. I[y, z] = 0 ( y2  z 2  2 yz )dt


y(0) = 1, y    0, z(0)
2

= – 1, z   0
2

P. T. O.