Ordinary Differential Equations (ODE)

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Ordinary Differential Equations (ODE)

Previous year Questions from 2018 to 1992

Ramanasri Institute

WEBSITE:

CONTACT: 8750706262/6363

1

2018

1.

Solve: y '' y x 2 e 2 x

[10 Marks]

2.

Solve: y ''' 6 y '' 1 2 y ' 8 y 1 2 e 2 x 2 7 e x

[10 Marks]

3.

(i) Find the Laplace transform of f (t ) 1 .

t

[10 Marks]

(ii) Find the Inverse Laplace transform of 5 s 2 3 s 1 6

( s 1)( s 2 )( s 3)

2

4.

Solve:

dy

dy y2 x y 0

dx

dx

5.

Solve: y '' 1 6 y 3 2 s e c 2 x

[13 Marks] [13 Marks]

6.

Solve: (1 x ) 2 y '' (1 x ) y ' y 4 c o s (lo g (1 x ))

7. Solve the initial value problem

[13 Marks]

y '' 5 y ' 4 y e 2 t ; y ( 0 ) 1 9 , y '( 0 ) 8

12

3

[13 Marks]

8.

Find and such that x y an integrating factor of ( 4 y 2 3 xy ) d x (3 xy 2 x 2 ) d y 0 and solve the

equation.

[12 Marks]

9.

Find f ( y ) such that ( 2 xe y 3 y 2 ) d y (3 x 2 f ( y )) d x 0 is exact and hence solve.

[12 Marks]

2017

10. Find the differential equation representing the entire circle in the xy - plane.

[10 Marks]

11. Solve the following simultaneous liner differential equations: ( D 1) y z e x and ( D 1) z y e x

where y and z are functions of independent variable x and D d . dx

[8 Marks]

12. If the growth rate of the population of bacteria at time t is proportional to the amount present at

the time and population doubles in one week, then how much bacteria's can be expected after 4

weeks?

[8 Marks]

13. Consider the differential equation xyp 2 ( x 2 y 2 1) p xy 0 where p d y substituting u x 2 and dx

v y 2 reduce the equation to Clairaut's form in terms of u , v and p ' d v hence or otherwise solve du

the equation.

[10 Marks]

14. Solve the following initial value differential equations 2 0 y " 4 y ' y 0, y (0 ) 3 .2, y '(0 ) 0 . [7 Marks]

15.

Solve the differential equation:

d2y x

dy

4x3y

8 x3 sin( x 2 )

dx2 dx

16. Solve that following differential equation using method of variation of parameters

[9 Marks]

. d 2 y d y 2 y 4 4 7 6 x 4 8 x 2

dx2 dx

[8 Marks]

17. Solve the following initial value problem using Laplace transform: d 2 y 9 y r ( x ), y (0 ) 0, y '(0 ) 4 dx2

where . r ( x )

8 sin x if 0 x

0

if x

[17 Marks]

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2016

18. Find a particular integral of d 2 y y e x / 2 s in x 3

dx 2

2

19. Show that the family of parabolas y 2 4 cx 4 c 2 is self orthogonal.

[10 marks] [10 marks]

20. Solve { y (1 x t a n x ) x 2 c o s x }d x x d y 0 21. Using the method of variation of parameter solve the differential equation

[10 marks]

( D 2 2 D 1) y e x log(x ),

d

D

dx

[15 marks]

22. Find the general solution of the equation x 2 d 3 y 4 x d 2 y 6 d y 4

dx3

dx 2

dx

[15 marks]

23. Using Laplace transformation solves the following: y '' 2 y ' 8 y 0 , y ( 0 ) 3 , y '( 0 ) 6 [10 marks]

2015

24. Solve the differential equation: x c o s x d y y ( x s in x c o s x ) 1

dx

25. Solve the differential equation: ( 2 xy 4 e y 2 xy 3 y ) d x ( x 2 y 4 e y x 2 y 2 3 x ) d y 0

26. Find the constant a so that ( x y ) a is the integrating factor of

( 4 x 2 2 xy 6 y ) d x ( 2 x 2 9 y 3 x ) d y 0 and hence solve the differential equation

[10 Marks] [10 Marks]

[12 Marks]

27. (i) (ii)

Obtain Laplace Inverse transform of

ln

1

1 s2

s2

s

25

e

5

s

Using Laplace transform, solve y " y t , y ( 0 ) 1, y '( 0 ) 2

[6+6=12 Marks]

28. Solve the differential equation x p y p 2 where p d y

dx

29.

Solve

x4

d4y dx4

6 x3

d3y dx3

4x2

d2y dx2

2x

dy dx

4y

x2

2 cos(log e

x)

[13 Marks] [13 Marks]

2014

30. Justify that a differential equation of the form: y x f ( x 2 y 2 ) d x y f ( x 2 y 2 ) x d y 0 where

f ( x 2 y 2 ) is an arbitrary function of ( x 2 y 2 ), is not an exact differential equation and 1 is

x2 y2

an integrating factor for it. Hence solve this differential equation for f ( x 2 y 2 ) ( x 2 y 2 ) 2

[10 Marks]

31. Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of

tangency

[10 Marks]

32. Solve by the method of variation of parameters: d y 5 y s in x

dx

(10 Marks]

33.

Solve the differential equation:

x3

d3y dx3

3x2

d2y dx2

x

dy dx

8y

65 cos log e

x

[20 Marks]

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34.

Solve the following differential equation:

x

d2y

dy

2 x 1

(x

2) y

x

2 e 2 x , when e x

is a

dx2

dx

solution to its corresponding homogeneous differential equation.

[15 Marks]

35. Find the sufficient condition for the differential equation M x , y d x N x , y d y 0, to have an

integrating factor as a function of ( x y ) . What will be the integrating factor in that case? Hence

find the integrating factor for the differential equation of ( x 2 xy )d x ( y 2 xy )d y 0 and solve it. [15 Marks]

36. Solve the initial value problem d 2 y y 8 e 2 t s in t , y ( 0 ) 0 , y '( 0 ) 0 by using Laplace transform.

dt2

[20 Marks]

2013

37. If y is a function of x , such that the differential coefficient d y is equal to co s x y s in x y .

dx

Find out a relation between x and y, which is free from any derivative / differential. [10 Marks]

38. Obtain the equation of the orthogonal trajectory of the family of curves represented by

r n a s in n , ( r , ) being the plane polar coordinates.

[10 Marks]

39. Solve the differential equation ( 5 x 3 1 2 x 2 6 y 2 )d x 6 x y d y 0

[15 Marks]

40. Using the method of variation of parameters, solve the differential equation d 2 y a 2 y s e c a x

dx 2

[15 Marks]

41. Find the general solution of the equation x 2 d 2 y x d y y ln x s in (ln x )

dx 2

dx

[15 Marks]

42. By using Laplace transform method, solve the differential equation ( D 2 n 2 )x a s in ( n t ),

D 2 d 2 subject to the initial conditions x 0 and d x 0 , at t 0 , in which a , n and are

dt2

dt

constants.

[15 Marks]

2012

43. Solve d y

2 x y e ( x / y )2

dx

y 2 (1 e ( x / y )2 ) 2 x 2 e ( x / y )2

[12 Marks]

44. Find the orthogonal trajectory of the family of curves x 2 y 2 a x

[12 Marks]

45. Using Laplace transforms, solve the initial value problem y '' 2 y ' y e t , y ( 0 ) 1 , y '( 0 ) 1 [12 Marks]

46. Show that the differential equation ( 2 x y lo g y )d x ( x 2 y 2 y 2 1 )d y 0 is not exact. Find an

integrating factor and hence, the solution of the equation

[20 Marks]

47. Find the general solution of the equation y ''' y '' 1 2 x 2 6 x

[20 Marks]

48. Solve the ordinary differential equation x ( x 1 ) y '' ( 2 x 1 ) y ' 2 y x 2 ( 2 x 3 )

[20 Marks]

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2011

49.

Obtain the solution of the ordinary differential equation

dy

4x

y

1

2

,

if

y(0)

1

[10 Marks]

dx

50. Determine the orthogonal trajectory of a family of curves represented by the polar equation

r a (1 c o s ), ( r , ) being the plane polar coordinates of any point.

[10 Marks]

51.

Obtain Clairaut's form of the differential equation

x

dy dx

y

y

dy dx

x

a2

dy dx

. Also find its

general solution

[15 Marks]

52. Obtain the general solution of the second order ordinary differential equation

y '' 2 y ' 2 y x e x c o s x , where dashes denote derivatives w.r.t. x

[15 Marks]

53. Using the method of variation of parameters, solve the second order differential equation

d2y 4 y tan 2x

dx 2

54. Use Laplace transform method to solve the following initial value problem:

[15 Marks]

d 2x

dx 2

x

et ,

x (0 )

2 and

dy

1

dt2

dt

dt t0

[15 Marks]

2010

55. Consider the differential equation y ' x , x 0 where is a constant. Show that

(i)

If ( x ) is any solution and ( x ) ( x )e x , then ( x ) is a constant;

(ii) If 0 , then every solution tends to zero as x

[12 Marks]

56. Show that the differential equation (3 y 2 x ) 2 y ( y 2 3) y ' 0 admits an integrating factor which

is a function of ( x y 2 ) . Hence solve the equation

[12 Marks]

57.

Verify that 1

2

(Mx

N y )d

lo g e ( x y )

1 2

(Mx

N y )d

lo g e ( x

/

y )

M dx

Ndy

. Hence show

that-

(i) If the differential equation M d x N d y 0 is homogeneous, then ( M x N y ) is an

integrating factor unless M x N y 0 ;

(ii) If the differential equation M d x N d y 0 is not exact but is of the form

f (xy)ydx 1

f (xy)xdy 2

0

then ( M x N y ) 1 is an integrating factor unless M x N y 0 ;

[20 Marks]

58. Use the method of undermined coefficients to find the particular solutions of

y '' y s in x (1 x 2 )e x and hence find its general solution.

[20 Marks]

2009

59. Find the Wronskian of the set of functions: 3 x 3 , 3 x 3 on the interval [ 1, 1 ] and determine

whether the set is linearly dependent on[ 1, 1 ]

[12 Marks]

60. Find the differential equation of the family of circles in the x y- plane passing through ( 1, 1 ) and

(1 , 1 )

[20 Marks]

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61.

Find the inverse Laplace transform of

F

(s)

1n

s s

1 s

62. Solve : d y

y2(x y)

, y(0) 1

dx 3xy2 x 2y 4 y3

2008

[20 Marks] [20 Marks]

63. Solve the differential equation y d x ( x x 3 y 2 )d y 0 64. Use the method of variation of parameters to find the general solution of

x 2 y '' 4 xy ' 6 y x 4 sin x

[12 Marks] [12 Marks]

65. Using Laplace transform, solve the initial value problem y '' 3 y ' 2 y 4 t e 3t , y ( 0 ) 1 ,

y '(0 ) 1

[15 Marks]

66. Solve the differential equation x 3 y '' 3 x 2 y ' x y s in (ln x ) 1

[15 Marks]

67. Solve the equation y 2 x p y p 2 0 , where p d y

dx

[15 Marks]

2007

68. Solve the ordinary differential equation c o s 3 x d y 3 y s in 3 x 1 s in 6 x s in 2 3 x , 0 x

dx

2

2

[12 Marks]

69. Find the solution of the equation d y x y 2 d x 4 x d x

y

[12 Marks]

1

70.

Determine the general and singular solutions of the equation

y

x

dy dx

a

dy dx

1

dy

d

x

2

2

,

a

being a constant.

[15 Marks]

71.

Obtain the general solution of

[D 3

6D2

12D

8]y

1

2

e

2x

9 4

ex

,

where

D

dy dx

[15 Marks]

72. Solve the equation 2 x 2 d 2 y 3 x d y 3 y x 3

dx 2

dx

[15 Marks]

73. Use the method of variation of parameters to find the general solution of the equation

d2y 3 dy 2 y 2ex

dx 2

dx

[15 Marks]

2006

74. Find the family of curves whose tangents form an angle with the hyperbolas x y c , c 0

4

[12 Marks]

1

75. Solve the differential equation x y 2 e x 3 d x x 2 y d y 0

[12 Marks]

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76.

Solve: (1 y 2 ) ( x e ta n 1 y ) d y 0

dx

[15 Marks]

77. Solve the equation x 2 p 2 p y 2 x y y 2 0 using the substitution y u and x y v and find its

singular solution, where p d y

dx

78.

Solve the differential equation x 2

d3y dx 3

2x

d2y dx 2

2

y x

10 1

1 x2

[15 Marks] [15 Marks]

79. Solve the differential equation ( D 2 2 D 2 ) y e x t a n x , D d y by the method of variation of

dx

parameters.

[15 Marks]

2005

80. Find the orthogonal trajectory of the family of co-axial circles x 2 y 2 2 g x c 0 , where g is

the parameter.

[12 Marks]

81.

Solve: x y d y ( x 2 y 2 x 2 y 2 1 )

dx

[12 Marks]

82.

Solve the differential equation:

(

x

1)4 D 3

2(x

1)3 D 2

(x

1)2 D

(x

1

)

y

1 (x 1)

[15 Marks]

83. Solve the differential equation: ( x 2 y 2 )(1 p )2 2 ( x y )(1 p )( x y p ) ( x y p )2 0 where

p d y , by reducing it to Clairaut's form by using suitable substitution.

dx

[15 Marks]

84. Solve the differential equation (s in x x c o s x ) y '' x s in x y ' y s in x 0 given that y s in x is a

solution of this equation.

[15 Marks]

85. Solve the differential equation x 2 y '' 2 x y ' 2 y x lo g x , x 0 by variation of parameters

[15 Marks]

2004

86. Find the solution of the following differential equation d y y c o s x 1 s in 2 x

dx

2

87. Solve: y ( x y 2 x 2 y 2 )d x x ( x y x 2 y 2 )d y 0

[12 Marks] [12 Marks]

88. Solve: ( D 4 4 D 2 5 ) y e x ( x c o s x )

[15 Marks]

89. Reduce the equation ( p x y )( p y x ) 2 p , where p d y to Clairaut's equation and hence solve

dx

it.

[15 Marks]

90.

Solve:

(x

d2y 2)

(2x

5)

dy

2y

(x

1)e x

dx 2

dx

[15 Marks]

91. Solve the following differential equation: (1 x 2 ) d 2 y 4 x d y (1 x 2 ) y x

dx 2

dx

[15 Marks]

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2003

92. Show that the orthogonal trajectory of a system of con-focals ellipses is self orthogonal [12 Marks]

93. Solve: x d y y lo g y x y e x

dx

[12 Marks]

94. Solve ( D 5 D ) 4 ( e x c o s x x 3 ), where D d y .

dx

[15 Marks]

95. Solve the differential equation ( p x 2 y 2 )( p x y ) ( P 1 )2 , where p d y , by reducing it to

dx

Clairaut's form using suitable substitutions

[15 Marks]

96. Solve (1 x 2 ) y '' (1 x ) y ' y s in 2 lo g (1 x )

[15 Marks]

97. Solve the differential equation x 2 y '' 4 x y ' 6 y x 4 s e c 2 x by variation of parameters. [15 Marks]

2002

98. Solve : x d y 3 y x 3 y 2

dx

[12 Marks]

99. Find the values of for which all solutions of x 2 d 2 y 3 x d y y 0 tend to zero as x .

dx 2

dx

[12 Marks]

100. Find the value of constant such that the following differential equation becomes exact.

( 2 x e y 3 y 2 ) d y (3 x 2 e y ) 0 . Further, for this value of , solve the equation.

dx

[15 Marks]

101. Solve : d y x y 4

dx x y 6

[15 Marks]

102. Using the method of variation of parameters, find the solutions of d 2 y 2 d y y x e x s in x with

d 2x

dx

y(0 )

0

and

dy

d

x

x 0

0

[15 Marks]

103. Solve : ( D 1 )( D 2 2 D 2 ) y e x where D d y

dx

[15 Marks]

2001

104.

105. 106. 107. 108.

A continuous function

y (t ) satisfies the differential equation d y

1

e1t ,

0 t 1 If

d x 2 2 t 3 t 2 , 1 t 5

y ( 0 ) e find y ( 2 )

[12 Marks]

Solve :

x2

d2y dx 2

x

dy dx

3y

x2

log e

x

[12 Marks]

Solve :

dy dx

y

x

log e

y

y(log y )2 e x2

Find the general solution of a y p 2 ( 2 x b ) p y 0 , a 0

[15 Marks] [15 Marks]

Solve: ( D 2 1 )2 y 2 4 x c o s x given that y D y D 2 y 0 and D 3 y 1 2 when x 0 [15 Marks]

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