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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD I YEAR B.TECH. Ist MIDE TERM EXAMINATION MATHEMATICS-I - 07A1BS02 UNIT-1 1. 2.
3.
2
3
Eliminate C from the equation y = ( x − C ) .
[5M]
Find the differential equation of all circles which pass through the origin and whose centers are on the x axis. [5M] x −x Eliminate A and B from the equation, xy = Ae +B e +x 2 . [5M]
4.
Find the differential equation of all parabolas with latus rectum ‘4a’ and axis parallel to the x-axis. [5M]
5.
Find the differential equation of all circles whose radius is `a’.
6.
Form the differential equation from y = e (A cos x + B sin x).
7.
Form the differential equation by elimiting the arbitrary constant y =
8.
[5M]
x
Solve the differential equation
xy
dy dx
=1 + x+ y +xy.
Solve the differential equation
10.
2 Solve the differential equation ( x + y + 1)
11.
Solve the differential equation y
dy =1. dx
12.
Solve the differential equation
13.
Solve the differential equation e
14
Solve the differential equation
15.
Solve the differential equation y − x
= (4x + y+ 1)2.
x− y
dx+e
ydx − xdy ( x − y)
2
y−x
=
2 1 − x2
dy dy = x+ y . dx dx
[5M]
[5M] [5M]
dy=0. dx
.[5M]
[5M]
1 − x 2 dy+x 1 − y 2 dx =0.
dy dx
a+x x2 + 1
[5M]
dy 2 x −3 y + x 2 e−3 y . dx = e
9.
[5M]
[5M] .
[5M]
[5M] Contd…2
::2:: 1
16.
Solve the differential equation xy + y + 4 = 0 .
17.
Solve the differential equation
18. 19. 20.
21.
22.
[5M]
1 − y 2 dx = (sin −1 y − x)dy .
[5M]
dy + 2 xy =1. dx dy + y =2 log x. Solve the differential equation x log x dx y dy x Solve the differential equation e + 1 = e . dx 2 Solve the differential equation ( x − 1)
[5M] [5M] [5M]
dy 2 Solve the differential equation x dx + y= y log x. dy 2 Solve the differential equation dx + y tan x = y sec x.
22.
Solve the differential equation x
23.
Solve the differential equation
24.
Solve the differential equation
[5M]
[5M]
dy + y = log x . dx
x2
[5M]
dy = ey − x . dx
[5M]
( Secx tan x tan y − e x )dx + sec x sec 2 ydy = 0 . 2
[5M]
2
25.
Solve the differential equation ydx − xdy = a( x + y )dx .
[5M]
26.
Solve the differential equation
y (2 xy + e x )dx = e x dy .
[5M]
3
dy ) dx
27.
Solve the differential equation (x+2y
28.
Prove that the system of Parabolas y = 4a ( x + a ) is self orthogonal.
29.
Find the orthogonal Trajectories of the family of curves r = a cos 2θ [5M]
30.
The number N of bacteria in a culture grew at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What was the value of N after
=y.
[5M]
2
2
1 1 hrs! . 2
[5M]
2
[5M] Contd…..3
::3:: 31.
If the air is maintained at 300 C and the temperature of the body cools from 800 C to 600 C in 12 minutes, find the temperature of the body after 24 minutes. [5M]
32.
The rate at which bacteria multiply is proportional to the instantaneous N numbers present. If the original number doubles in 2hrs ? When it will be tripled? [5M]
33.
The temperature of the body drops from 100 C to 75 C in ten minutes when the
0
0
0
surrounding air is 20 C. What will be it’s temperature after v half an hour. When 0
will the temperature be 25 C.
[5M]
34.
Find the orthogonal trajectories of y2 = 4ax.
[5M]
35.
The temperature of the body drops from 70C to 40 C in ten minutes when the surrounding air is 15 C. wat will be it’s temperature after 30 minutes. [5M]
36.
Find the orthogonal trajectories of r=a(1- cos
37.
Find the orthogonal trajectories of ay2 = x .
38.
the rate at which a certain substance decomposes in a certain solution at any instant proportional to the amount of it present in the solution at that instant. Initially there are 27 grams and three hours later, it is found to be 8 gms. How much will be left after one hour more? [5M]
39.
A bacterial culture growing from 100 to 400 gms in 10 hrs. How much will be present after 3 hrs? [5M]
40.
The temperature of the body drops from 120 C to 80
θ ).
[5M]
3
0
surrounding air is 25
0
[5M]
0
C in ten minutes when the 0
C. When will be it’s temperature 30 C? ---------
[5M]
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD I YEAR B.TECH. Ist MIDE TERM EXAMINATION MATHEMATICS-I - 07A1BS02 UNIT-II 1. 2.
3
Solve the differential equation ( D + 27) y = 0 . 2
[5M]
2
Solve the differential equation ( D − 2 D + 4) y = 0
[5M]
4
3.
Solve the differential equation ( D − 16) y = 0
4.
3 2 Solve the differential equation ( D − 1) ( D + 2 D + 2) y = 0
5.
Solve the differential equation y ′′ − y ′ − 2 y = 3.e
[5M] 2
2x
Given that y (0) = 0,
y ′(0) = −2 6. 7. 8.
[5M]
[5M]
3 x Solve the differential equation ( D − 1) y = e .
[5M]
2
Solve the differential equation ( D + 9) y = cos 3 x + sin 2 x . 2
Solve the differential equation ( D + 6 D + 9) y = e
d2y
−3 x
.
[5M] [5M]
− y = x sin x .
[5M]
9.
Solve the differential equation
10.
Solve the differential equation (D D
11.
Solve the differential equation (D -4D +4)y = cos x.
12.
Solve the differential equation ( D + 9) y = cos3x .
[5M]
13.
3 2 Solve the differential equation ( D − 3D − 2) y = x .
[5M]
14.
Solve the differential equation ( D + 2 D + D ) y = e
15.
2 2 Solve the differential equation ( D − 4) y = 2 cos x .
16. 17. 18. 19. 20.
dx 2
2
+4)y = sin 2x+4)y = sin 2x.
2
[5M]
2
3
2 +16)y = x . 2 2x Solve the differential equation (4D +4D-3) y= e . 2 2x Solve the differential equation (D +2D-3) y= e . 2 2 Solve the differential equation (D-2) y= x . 2 2x Solve the differential equation (D -4D+4)y e . Solve the differential equation (D
2
2
[5M]
2x
.
[5M] [5M] [5M] [5M] [5M] [5M] [5M] Contd….2
::2::
2
3
21.
Solve the differential equation (D
22.
Solve the differential equation ( D + 1) y = 0 , Given that y = 2, y ' = −2 when x=0. [5M] 2 x [5M] Solve the differential equation (D -2D+1)y = e +2. 2 2x Solve the differential equation (D -8D +16)y=e . [5M] 2 3x . [5M] Solve the differential equation (D +9)y= e 2 x Solve the differential equation (D -2D+2) y = x e e . [5M] 2 Solve the differential equation (D -2D+$)y = cos x. [5M] 2 Solve the differential equation (D +3D+2)y=x. [5M] 2 [5M] Solve the differential equation (D +3D+6)y=x.
23. 24. 25. 26. 27. 28. 29.
x
39. 40.
Solve the differential equation (D 3
31. 32. 33. 34. 35. 36. 37. 38.
.
[5M]
2
Solve the differential equation (D e 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D Solve the differential equation (D 3 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D
30.
-9D+ 9)y = x
x
-2D+1)y =x e .
[5M]
-2D+!)y =cos2x. 2 -2D+4)y = x .
[5M]
-D+4)y= sin 2x. 3x +4D-1)y= e . -8)y=cosx.
[5M] [5M] [5M]
-2D)y=x.
[5M]
-3D+4)y=x e
2x
[5M]
.
x
-2D+1)y = x e . 4x
-2D+6)y=e . 2 4x -4D -D+4)y= e . --------
[5M] [5M] [5M] [5M]
3.
2
3
Eliminate C from the equation y = ( x − C ) .
[5M]
Find the differential equation of all circles which pass through the origin and whose centers are on the x axis. [5M] x −x Eliminate A and B from the equation, xy = Ae +B e +x 2 . [5M]
4.
Find the differential equation of all parabolas with latus rectum ‘4a’ and axis parallel to the x-axis. [5M]
5.
Find the differential equation of all circles whose radius is `a’.
6.
Form the differential equation from y = e (A cos x + B sin x).
7.
Form the differential equation by elimiting the arbitrary constant y =
8.
[5M]
x
Solve the differential equation
xy
dy dx
=1 + x+ y +xy.
Solve the differential equation
10.
2 Solve the differential equation ( x + y + 1)
11.
Solve the differential equation y
dy =1. dx
12.
Solve the differential equation
13.
Solve the differential equation e
14
Solve the differential equation
15.
Solve the differential equation y − x
= (4x + y+ 1)2.
x− y
dx+e
ydx − xdy ( x − y)
2
y−x
=
2 1 − x2
dy dy = x+ y . dx dx
[5M]
[5M] [5M]
dy=0. dx
.[5M]
[5M]
1 − x 2 dy+x 1 − y 2 dx =0.
dy dx
a+x x2 + 1
[5M]
dy 2 x −3 y + x 2 e−3 y . dx = e
9.
[5M]
[5M] .
[5M]
[5M] Contd…2
::2:: 1
16.
Solve the differential equation xy + y + 4 = 0 .
17.
Solve the differential equation
18. 19. 20.
21.
22.
[5M]
1 − y 2 dx = (sin −1 y − x)dy .
[5M]
dy + 2 xy =1. dx dy + y =2 log x. Solve the differential equation x log x dx y dy x Solve the differential equation e + 1 = e . dx 2 Solve the differential equation ( x − 1)
[5M] [5M] [5M]
dy 2 Solve the differential equation x dx + y= y log x. dy 2 Solve the differential equation dx + y tan x = y sec x.
22.
Solve the differential equation x
23.
Solve the differential equation
24.
Solve the differential equation
[5M]
[5M]
dy + y = log x . dx
x2
[5M]
dy = ey − x . dx
[5M]
( Secx tan x tan y − e x )dx + sec x sec 2 ydy = 0 . 2
[5M]
2
25.
Solve the differential equation ydx − xdy = a( x + y )dx .
[5M]
26.
Solve the differential equation
y (2 xy + e x )dx = e x dy .
[5M]
3
dy ) dx
27.
Solve the differential equation (x+2y
28.
Prove that the system of Parabolas y = 4a ( x + a ) is self orthogonal.
29.
Find the orthogonal Trajectories of the family of curves r = a cos 2θ [5M]
30.
The number N of bacteria in a culture grew at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What was the value of N after
=y.
[5M]
2
2
1 1 hrs! . 2
[5M]
2
[5M] Contd…..3
::3:: 31.
If the air is maintained at 300 C and the temperature of the body cools from 800 C to 600 C in 12 minutes, find the temperature of the body after 24 minutes. [5M]
32.
The rate at which bacteria multiply is proportional to the instantaneous N numbers present. If the original number doubles in 2hrs ? When it will be tripled? [5M]
33.
The temperature of the body drops from 100 C to 75 C in ten minutes when the
0
0
0
surrounding air is 20 C. What will be it’s temperature after v half an hour. When 0
will the temperature be 25 C.
[5M]
34.
Find the orthogonal trajectories of y2 = 4ax.
[5M]
35.
The temperature of the body drops from 70C to 40 C in ten minutes when the surrounding air is 15 C. wat will be it’s temperature after 30 minutes. [5M]
36.
Find the orthogonal trajectories of r=a(1- cos
37.
Find the orthogonal trajectories of ay2 = x .
38.
the rate at which a certain substance decomposes in a certain solution at any instant proportional to the amount of it present in the solution at that instant. Initially there are 27 grams and three hours later, it is found to be 8 gms. How much will be left after one hour more? [5M]
39.
A bacterial culture growing from 100 to 400 gms in 10 hrs. How much will be present after 3 hrs? [5M]
40.
The temperature of the body drops from 120 C to 80
θ ).
[5M]
3
0
surrounding air is 25
0
[5M]
0
C in ten minutes when the 0
C. When will be it’s temperature 30 C? ---------
[5M]
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD I YEAR B.TECH. Ist MIDE TERM EXAMINATION MATHEMATICS-I - 07A1BS02 UNIT-II 1. 2.
3
Solve the differential equation ( D + 27) y = 0 . 2
[5M]
2
Solve the differential equation ( D − 2 D + 4) y = 0
[5M]
4
3.
Solve the differential equation ( D − 16) y = 0
4.
3 2 Solve the differential equation ( D − 1) ( D + 2 D + 2) y = 0
5.
Solve the differential equation y ′′ − y ′ − 2 y = 3.e
[5M] 2
2x
Given that y (0) = 0,
y ′(0) = −2 6. 7. 8.
[5M]
[5M]
3 x Solve the differential equation ( D − 1) y = e .
[5M]
2
Solve the differential equation ( D + 9) y = cos 3 x + sin 2 x . 2
Solve the differential equation ( D + 6 D + 9) y = e
d2y
−3 x
.
[5M] [5M]
− y = x sin x .
[5M]
9.
Solve the differential equation
10.
Solve the differential equation (D D
11.
Solve the differential equation (D -4D +4)y = cos x.
12.
Solve the differential equation ( D + 9) y = cos3x .
[5M]
13.
3 2 Solve the differential equation ( D − 3D − 2) y = x .
[5M]
14.
Solve the differential equation ( D + 2 D + D ) y = e
15.
2 2 Solve the differential equation ( D − 4) y = 2 cos x .
16. 17. 18. 19. 20.
dx 2
2
+4)y = sin 2x+4)y = sin 2x.
2
[5M]
2
3
2 +16)y = x . 2 2x Solve the differential equation (4D +4D-3) y= e . 2 2x Solve the differential equation (D +2D-3) y= e . 2 2 Solve the differential equation (D-2) y= x . 2 2x Solve the differential equation (D -4D+4)y e . Solve the differential equation (D
2
2
[5M]
2x
.
[5M] [5M] [5M] [5M] [5M] [5M] [5M] Contd….2
::2::
2
3
21.
Solve the differential equation (D
22.
Solve the differential equation ( D + 1) y = 0 , Given that y = 2, y ' = −2 when x=0. [5M] 2 x [5M] Solve the differential equation (D -2D+1)y = e +2. 2 2x Solve the differential equation (D -8D +16)y=e . [5M] 2 3x . [5M] Solve the differential equation (D +9)y= e 2 x Solve the differential equation (D -2D+2) y = x e e . [5M] 2 Solve the differential equation (D -2D+$)y = cos x. [5M] 2 Solve the differential equation (D +3D+2)y=x. [5M] 2 [5M] Solve the differential equation (D +3D+6)y=x.
23. 24. 25. 26. 27. 28. 29.
x
39. 40.
Solve the differential equation (D 3
31. 32. 33. 34. 35. 36. 37. 38.
.
[5M]
2
Solve the differential equation (D e 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D Solve the differential equation (D 3 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D 2 Solve the differential equation (D
30.
-9D+ 9)y = x
x
-2D+1)y =x e .
[5M]
-2D+!)y =cos2x. 2 -2D+4)y = x .
[5M]
-D+4)y= sin 2x. 3x +4D-1)y= e . -8)y=cosx.
[5M] [5M] [5M]
-2D)y=x.
[5M]
-3D+4)y=x e
2x
[5M]
.
x
-2D+1)y = x e . 4x
-2D+6)y=e . 2 4x -4D -D+4)y= e . --------
[5M] [5M] [5M] [5M]
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