# RD SHARMA Solutions for Class 12-science Maths Chapter 22 - Differential Equations

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

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## 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
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Question 22
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Question 23
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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
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Question 3
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Question 5
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Question 7
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Question 27
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Question 28
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Question 29
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Question 30
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Question 31
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Question 32
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Question 33
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Question 34
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Question 41

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Question 59

dy = cos x (2 - y cosec x) dx

Solution 59

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Solution 63

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Question 13
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Question 22
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Question 27

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

## Chapter 22 - Differential Equations Exercise Ex. 22.2

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

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Question 19
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Question 20

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Question 21
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Question 22
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Question 27
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Question 28
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Question 29
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Question 30
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Question 31
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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
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Question 5
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Question 6
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Question 7
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Question 8
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Question 9
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Question 10

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Question 11
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Question 12

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

Solution 12

Question 13
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Question 14
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Question 17

Solution 17

Question 18
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Solution 25

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## Chapter 22 - Differential Equations Exercise Ex. 22.5

Question 1
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Question 2
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Question 3
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Question 5

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Question 7

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Question 8

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Question 9
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Question 11

Solve the following differential equation:

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

Solution 11

Question 12
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Question 13
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Question 14
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Question 15
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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
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Question 28
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Question 29
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Question 34
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Question 35
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Question 36
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Question 37

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Question 38

Solve the following differential equation:

Solution 38

Question 39
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Question 40
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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
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Question 44
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Question 48
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Question 64

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

Solution 64

Question 65
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Question 66
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Question 69
Solution 69
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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
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Question 9

Solution 9

Question 10
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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
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Question 8

Solution 8

Question 9
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Question 10
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Question 11
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Question 12

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Question 15
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Question 19
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Question 20
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Question 22
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Question 23
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Question 25
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Question 26

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Question 27

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Question 28
Solution 28
Question 29

Solution 29

Question 30

Solution 30

Question 31
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Question 32
Solution 32
Question 33
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Question 34

Solution 34

Question 35

Solve the following differential equation:

Solution 35

Question 36
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Question 37
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Question 38
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Question 39
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Question 40
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Question 41
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## 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

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. xy1- ln 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)

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

(a) xy =C

(b) x = Cy2

(c) y = Cx

(d) y = Cx2

Solution 52

Correct option:(c)

Question 53

Solution 53

Correct option: (c)

Question 54

The general solution of the differential equation exdy+(yex+2x) dx =0 is

(a) x ey+x2=C

(b) x ey+y2=C

(c) y ex+x2=C

(d) y ey+x2=C

Solution 54

Correct option: (c)

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