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RD Sharma Solution CLass 12 Mathematics Chapter 22 Differential Equations

Chapter 22 – Differential Equations Exercise Ex. 22.1

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Solution 4

Question 5
Solution 5
Question 6

Solution 6

Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17

Solution 17

Question 18
Solution 18
Question 19
Solution 19
Question 20

Solution 20

Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25

Solution 25

Question 26

Determine the order
and degree of the following differential equations. State also whether they
are linear or non-linear.

Solution 26

The order of a
differential equation is the order of the highest order derivative appearing
in the equation.

The degree of a
differential equation is the degree of the highest order derivative.

Consider the given
differential equation

In the above equation,
the order of the highest order derivative is 1.

So the differential
equation is of order 1.

In the above
differential equation, the power of the highest order derivative is 3.

Hence, it is a
differential equation of degree 3.

Since the degree of
the above differential equation is 3, more than one, it is a non-linear
differential equation.

Chapter 22 – Differential Equations Exercise Ex. 22.10

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31

Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solution 39
Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43

Solution 43

Question 44
Solution 44
Question 45
Solution 45
Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48
Question 49
Solution 49
Question 50
Solution 50
Question 51
Solution 51
Question 52
Solution 52
Question 53
Solution 53
Question 54
Solution 54
Question 55
Solution 55
Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

dy = cos x (2 – y cosec x) dx

Solution 59

Question 60
Solution 60
Question 61
Solution 61
Question 62
Solution 62
Question 63

Solution 63

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Solution 4

Question 5
Solution 5

Question 6
Solution 6

Question 7

Solution 7

Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12

Question 13
Solution 13
Question 14
Solution 14
Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Question 25
Solution 25
Question 26
Solution 26
Question 27

Solution 27

Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34

Chapter 22 – Differential Equations Exercise Ex. 22.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11

Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17

Form the differential equation having y = (sin-1x)2 + A cos -1 x + B, where A and B are arbitrary constants, as its general solution.

Solution 17

Question 18

Solution 18

Question 19
Solution 19
Question 20

Solution 20

Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32

Question 33
Solution 33

Chapter 22 – Differential Equations Exercise Ex. 22.3

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10

Solution 10

Question 11
Solution 11
Question 12

Show that y = ex(A cos x + B sin x) is the solution of the differential equation

Solution 12

Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17

Solution 17

Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23

Solution 23

Question 24
Solution 24
Question 25
Solution 25

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9

Chapter 22 – Differential Equations Exercise Ex. 22.5

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9
Solution 9
Question 10
Solution 10
Question 11

Solve the following differential equation:

(sin x + cos x)dy + (cos x – sin x) dx = 0

Solution 11

Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21

Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26

solve the following differential equation

Solution 26

Chapter 22 – Differential Equations Exercise Ex. 22.6

Question 1

Solve the following differential equation:

Solution 1

Question 2

Solve the following differential equation:

Solution 2

Question 3
Solution 3
Question 4

Solution 4

Chapter 22 – Differential Equations Exercise Ex. 22.7

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Solution 4

Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30

Solution 30

Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37

Solution 37

Question 38

Solve the following
differential equation:

Solution 38

Question 39
Solution 39
Question 40
Solution 40
Question 41

yex/y dx = (xex/y + y2) dy, y ¹ 0

Solution 41

Question 42

(1 + y2) tan-1 x dx + 2y (1 + x2)dy = 0

Solution 42

Question 43
Solution 43
Question 44
Solution 44
Question 45
Solution 45
Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48
Question 49
Solution 49
Question 50
Solution 50
Question 51
Solution 51
Question 52
Solution 52
Question 53
Solution 53
Question 54
Solution 54
Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58
Solution 58
Question 59
Solution 59
Question 60
Solution 60
Question 61
Solution 61
Question 62
Solution 62
Question 63
Solution 63
Question 64

Find the equation of a curve passing through the point (0,0) and whose differential equation is

Solution 64

Question 65
Solution 65

Question 66
Solution 66

Question 67
Solution 67
Question 68
Solution 68

Question 69
Solution 69

..
Question 70

Solution 70

Chapter 22 – Differential Equations Exercise Ex. 22.8

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9

Solution 9

Question 10
Solution 10
Question 11

Solve the following differential equation.

Solution 11

Chapter 22 – Differential Equations Exercise Ex. 22.9

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Solve the following differential equation:

Solution 4

Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8

Solution 8

Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12

Solution 12

Question 13
Solution 13

Question 14
Solution 14
Question 15
Solution 15

Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Question 25
Solution 25
Question 26

Solution 26

Question 27

Solution 27

Question 28
Solution 28
Question 29

Solution 29

Question 30

Solution 30

Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33

Question 34

Solution 34

Question 35

Solve the following differential equation:

Solution 35

Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solution 39
Question 40
Solution 40

Question 41
Solution 41

Question 42
Solution 42
Question 43

Solution 43

Question 44
Solution 44
Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64

Solution 64

Question 65

Solution 65

Question 66

Solution 66

Question 67

Solution 67

Question 68

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73

Question 74

Solution 74

Question 75

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79

Question 80

Solution 80

Question 81

Solution 81

Question 82

Solution 82

Question 83

Solution 83

Question 84

Solution 84

Question 85

Solution 85

Question 86

Solution 86

Question 87

Solution 87

Question 88

Solution 88

Question 89

Solution 89

Question 90

Solution 90

Question 91

Solution 91

Question 92

Solution 92

Question 93

Solution 93

Question 94

Solution 94

Question 95

Solution 95

Question 96

Solution 96

Question 97

Solution 97

Question 98

Solution 98

Question 99

Solution 99

Question 100

Solution 100

Question 101

Solution 101

Question 102

Solution 102

Question 103

Solution 103

Question 104

Solution 104

Question 105

Solution 105

Question 106

Solution 106

Question 107

Solution 107

Question 108

Solution 108

Question 109

Solution 109

Question 110

Solution 110

Question 111

Solution 111

Question 112

Solution 112

Chapter 22 – Differential Equations Exercise Ex. 22VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Write the differential equation
representing family of curves y = mx, where m is arbitrary constant.

Solution 20

Question 21

Solution 21

Chapter 22 – Differential Equations Exercise MCQ

Question 1

Mark the correct alternative in each of the following

Solution 1

Correct option: (c)

Question 2

1. log y =kx
2. y =kx
3. xy =k
4. y = k log x
Solution 2

Correct option: (b)

Question 3

1. Sin x
2. Sec x
3. Tan x
4. Cos x
Solution 3

Correct option: (b)

Question 4

1. ½
2. 2
3. 3
4. 4
Solution 4

Correct option:(b)

Degree is the power of highest order derivative.

Highest order is 2 and its power is 2.

Hence, degree of differential equation is 2.

Question 5

1. 4
2. 2
3. 5
4. 10

Solution 5

NOTE: Answer not matching with back answer.

Question 6

1. x + y sin x =C
2. x + y cos x = C
3. y+ x ( sin x + cos x) = C
4. y sin x = x + C
Solution 6

Correct option:(d)

Question 7

The different equation obtained on eliminating A and B from y = A cos ωt + B sin ωt ,is

1. Y+ y =0
2. Y”- ω2y=0
3. Y”= -ω2 y
4. Y + y =0
Solution 7

Correct option: (c)

Question 8

1. x2 = y
2. y2 = x
3. x2 = 2y
4. y2 = 2x

Solution 8

Correct option: (a)

Question 9

The order of the different equation whose general solution is given by y =c1 cos (2x +c2) – (c3 +c4) ax+c5 + c6  sin (x -c7) is

1. 3
2. 4
3. 5
4. 2
Solution 9

Correct option: (c)

Here, constants are c1, c2, c3, c4, c5, c6.

But c3+c4 is also constant. Hence, total 5 constants.

Question 10

1. a = b
2. a = -b
3. a =-2b
4. a =2b
Solution 10

Correct option: (b)

Question 11

Solution 11

Correct option: (a)

Question 12

Solution 12

Correct option: (a)

Question 13

1. 1
2. 2
3. 3
4. 4
Solution 13

Correct option: (a)

Differential equation contains only one constant hence,

Order of differential equation is 1.

Question 14

The solution of the differential equation y1 y3=y22 is

1. x= C1eC2y +C3
2. y= C1 eC2x +C3
3. 2x= C1 eC2y +C3
4. None of these
Solution 14

Correct option: (b)

Question 15

Solution 15

Correct option: (b)

Question 16

Solution 16

Correct option: (d)

Question 17

Solution 17

Correct option: (a)

Question 18

1. x( y + cos x) = sin x +C
2. x( y – cos x) = sin x +C
3. x( y + cos x) = cos x +C
4. None of these
Solution 18

Correct option: (a)

Question 19

The equation of the curve satisfying the differential equation y(x+y3) dx = x(y3-x) dy and passing through the point (1,1) is

1. y3-2x+3x2y =0
2. y3+2x+3x2y =0
3. y3+2x-3x2y =0
4. None of these
Solution 19

Correct option: (c)

Question 20

1. Circles
2. Straight lines
3. Ellipses
4. Parabolas
Solution 20

Correct option: (d)

Question 21

Solution 21

Correct option: (b)

Question 22

The different equation satisfied by ax2+by2=1 is

a. xyy2+y12+yy1=0

b. xyy2+xy12-yy1=0

c. xyy2-xy12+yy1=0

d. none of these

Solution 22

Correct option: (b)

Question 23

The different equation which represents the family of curves y = eCx is

1. y1= C2y
2. xy1ln y =0
3. x ln y = yy1
4. y ln y = xy1
Solution 23

Correct option: (d)

Note: log is considered same as ln.

Question 24

1. u = log x
2. u = ez
3. u = (log z)-1
4. u = (log z) 2

Solution 24

Correct option: (c)

Question 25

Solution 25

Correct option:(a)

Question 26

1. m = 3, n = 3
2. m = 3, n =2
3. m = 3, n =5
4. m =3, n =1

Solution 26

Correct option: (b)

Question 27

Solution 27

Correct option: (d)

Question 28

Solution 28

Correct option: (d)

Question 29

The family of curves in which the subtangent at any point of a curve is double the abscisae, is given by

1. x =Cy2
2. y =Cx2
3. x2 =Cy2
4. y =Cx
Solution 29

Correct option: (a)

Question 30

The solution of the differential equation x dx +y dy =x2y dy -y2 x dx , is

1. x2-1 = C (1+y2)
2. x2+1=C (1-y2)
3. x3-1=C (1+y3)
4. x3+1=C (1-y3)
Solution 30

Correct option:(a)

Question 31

Solution 31

Correct option:

Question 32

Solution 32

Correct option: (b)

Question 33

1. k =0
2. k > 0
3. k < 0
4. none of these
Solution 33

Correct option:(c)

Question 34

1. tan-1  x-tan-1 y = tan -1 C
2. tan-1  y-tan-1 x = tan -1 C
3. tan-1  y ± tan-1 x = tan C
4. tan-1  y +tan-1 x = tan -1 C
Solution 34

Correct option: (d)

Question 35

Solution 35

Correct option: (b)

Question 36

Solution 36

Correct option:(d)

Question 37

1. p < q
2. p = q
3. p > q
4. none of these
Solution 37

Correct option: (c)

Note: Answer not matching with back answer.

Question 38

1. x
2. ex
3. log x
4. log (log x)
Solution 38

Correct option: (c)

Question 39

1. sec x + tan x
2. log (sec x+ tan x)
3. esec x
4. sec x
Solution 39

Correct option: (a)

Question 40

(a) cos x

(b) tan x

(c) sec x

(d) sin x

Solution 40

Correct option: (c)

Question 41

(a) 3

(b) 2

(c) 1

(d) Not defined

Solution 41

Correct option: (d)

Highest order derivative is 2 but equation cannot be expressed as a polynomial in differential equation.

Hence, it is not defined.

Question 42

(a) 2

(b) 1

(c) 0

(d) Not defined

Solution 42

Correct option:(a)

Highest order of the derivative is 2.

Question 43

The number of arbitrary constants in the general solution of differential equation of fourth order is

(a) 0

(b) 2

(c) 3

(d) 4

Solution 43

Correct option: (d)

In the general solution of differential equation of order n has n number of arbitrary constants.

Question 44

The number of arbitrary constants in the particular solution of a differential equation of third order is

(a) 3

(b) 2

(c) 1

(d) 0

Solution 44

Correct option: (d)

The number of arbitrary constants in the particular solution of a differential equation of third order is always zer0.

Question 45

Solution 45

Correct option: (b)

Question 46

Which of the following differential equation has y = x as one of its particular solution?

Solution 46

Correct option: (c)

Question 47

(a) ex +e-y =C

(b) ex + ey= C

(c) e-x +ey =C

(d) e-x +e-y =C

Solution 47

Correct option: (a)

Question 48

(a) y = vx

(b) v = yx

(c) x = vy

(d) x = v

Solution 48

Correct option: (c)

Question 49

Which of the following is a homogeneous differential equation?

(a) (4x+6y+5) dy-(3y +2x+4) dx =0

(b) xy dx -(x3+y3)dy =0

(c) (x3+2y2)dx+2xy dy =0

(d) y2 dx+(x2-xy-y2) dy =0

Solution 49

Correct option: (d)

Question 50

1. e-x
2. e-y
3. 1/x
4. x
Solution 50

Correct option: (c)

Question 51

Solution 51

Correct option:(d)

Question 52