04-15-2011, 08:03 AM
full papers look attachment
Question No: 1 Marks: 2
The differential equation secydy sin(xy) sin(x
dx
+ − = + )y is
Separable
Question No: 2 Marks: 2
The integrating factor of the differential equation(x2 1)dy 2xy 1
dx
+ + = is
x2 +1
Question No: 3 Marks: 2
The form of the particular solution for the differential equation
y′′−y=x4
y =Ax4+Ax3+Ax2+Ax+A p 4 3 2 1 0
Question No: 4 Marks:2
Determine which of the given functions are linearly independent.
( ) ( ) 1 , , ( ) 2 1 2 2 f x= +xf x=xf x=x
Question No: 5 Marks:2
The differential operator that annihilates 10x3 − 2x is:
D4
Question No: 6 Marks:10
Solve the following differential equation by using an appropriate substitution.
dy y x
dx x y
= +
Solution
2 2
2 2 2
2
2
2
2
, ,
1
1 1
1 ln ln
2
2ln
dy y x
dx x y
dy y x
dx xy
Homogeneous equation so put y vx dy v x dv
dx dx
v xdv v x x
dx x v
v xdv v
dx v
x dv vdv dx
dx v x
vdv dx v x C
x
y xC
x
= +
+
⇒ =
⇒ =
+
⇒ + =
⇒ + = +
⇒ = ⇒ =
⇒ = ⇒ = +
⇒ =
∫ ∫
= +
Question No: 7 Marks:10
The population of a town grows at a rate proportional to the population at any time. Its initial population of
500 increases by 15% in 10 years. What will be the population in 30 years?
Solution:
Let P(t) be the population at any time t, then rate of grows will be
Question No: 1 Marks: 2
The differential equation secydy sin(xy) sin(x
dx
+ − = + )y is
Separable
Question No: 2 Marks: 2
The integrating factor of the differential equation(x2 1)dy 2xy 1
dx
+ + = is
x2 +1
Question No: 3 Marks: 2
The form of the particular solution for the differential equation
y′′−y=x4
y =Ax4+Ax3+Ax2+Ax+A p 4 3 2 1 0
Question No: 4 Marks:2
Determine which of the given functions are linearly independent.
( ) ( ) 1 , , ( ) 2 1 2 2 f x= +xf x=xf x=x
Question No: 5 Marks:2
The differential operator that annihilates 10x3 − 2x is:
D4
Question No: 6 Marks:10
Solve the following differential equation by using an appropriate substitution.
dy y x
dx x y
= +
Solution
2 2
2 2 2
2
2
2
2
, ,
1
1 1
1 ln ln
2
2ln
dy y x
dx x y
dy y x
dx xy
Homogeneous equation so put y vx dy v x dv
dx dx
v xdv v x x
dx x v
v xdv v
dx v
x dv vdv dx
dx v x
vdv dx v x C
x
y xC
x
= +
+
⇒ =
⇒ =
+
⇒ + =
⇒ + = +
⇒ = ⇒ =
⇒ = ⇒ = +
⇒ =
∫ ∫
= +
Question No: 7 Marks:10
The population of a town grows at a rate proportional to the population at any time. Its initial population of
500 increases by 15% in 10 years. What will be the population in 30 years?
Solution:
Let P(t) be the population at any time t, then rate of grows will be