# 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

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

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

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

## Chapter 22 - Differential Equations Exercise Ex. 22.11

## Chapter 22 - Differential Equations Exercise Ex. 22.2

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

## Chapter 22 - Differential Equations Exercise Ex. 22.3

Show that y = e^{x}(A cos x + B sin x) is the solution of the differential equation

## Chapter 22 - Differential Equations Exercise Ex. 22.4

## Chapter 22 - Differential Equations Exercise Ex. 22.5

Solve the following differential equation:

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

solve the following differential equation

## Chapter 22 - Differential Equations Exercise Ex. 22.6

Solve the following differential equation:

Solve the following differential equation:

## Chapter 22 - Differential Equations Exercise Ex. 22.7

Solve the following differential equation:

ye^{x}^{/y} dx = (xe^{x}^{/y} + y^{2}) dy, y ¹ 0

(1 + y^{2}) tan^{-1} x dx + 2y (1 + x^{2})dy = 0

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

## Chapter 22 - Differential Equations Exercise Ex. 22.8

Solve the following differential equation.

## Chapter 22 - Differential Equations Exercise Ex. 22.9

Solve the following differential equation:

Solve the following differential equation:

## Chapter 22 - Differential Equations Exercise Ex. 22RE

## Chapter 22 - Differential Equations Exercise Ex. 22VSAQ

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

## Chapter 22 - Differential Equations Exercise MCQ

Mark the correct alternative in each of the following

Correct option: (c)

- log y =kx
- y =kx
- xy =k
- y = k log x

Correct option: (b)

- Sin x
- Sec x
- Tan x
- Cos x

Correct option: (b)

- ½
- 2
- 3
- 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.

- 4
- 2
- 5
- 10

NOTE: Answer not matching with back answer.

- x + y sin x =C
- x + y cos x = C
- y+ x ( sin x + cos x) = C
- y sin x = x + C

Correct option:(d)

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

- Y
^{''}+ y^{'}=0 - Y''- ω
^{2}y=0 - Y''= -ω
^{2}y - Y
^{''}+ y =0

Correct option: (c)

- x
^{2 }= y - y
^{2 }= x - x
^{2}= 2y - y
^{2 }= 2x

Correct option: (a)

The order of the different equation whose general solution is given by y =c_{1 }cos (2x +c_{2}) - (c_{3} +c_{4}) a^{x+c}_{5} + c_{6 } sin (x -c_{7}) is

- 3
- 4
- 5
- 2

Correct option: (c)

Here, constants are c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}.

But c_{3}+c_{4} is also constant. Hence, total 5 constants.

- a = b
- a = -b
- a =-2b
- a =2b

Correct option: (b)

Correct option: (a)

Correct option: (a)

- 1
- 2
- 3
- 4

Correct option: (a)

Differential equation contains only one constant hence,

Order of differential equation is 1.

The solution of the differential equation y_{1} y_{3}=y_{2}^{2} is

- x= C
_{1}e^{C}_{2}^{y}+C_{3} - y= C
_{1}e^{C}_{2}^{x}+C_{3} - 2x= C
_{1}e^{C}_{2}^{y}+C_{3} - None of these

Correct option: (b)

Correct option: (b)

Correct option: (d)

Correct option: (a)

- x( y + cos x) = sin x +C
- x( y - cos x) = sin x +C
- x( y + cos x) = cos x +C
- None of these

Correct option: (a)

The equation of the curve satisfying the differential equation y(x+y^{3}) dx = x(y^{3}-x) dy and passing through the point (1,1) is

- y
^{3}-2x+3x^{2}y =0 - y
^{3}+2x+3x^{2}y =0 - y
^{3}+2x-3x^{2}y =0 - None of these

Correct option: (c)

- Circles
- Straight lines
- Ellipses
- Parabolas

Correct option: (d)

Correct option: (b)

The different equation satisfied by ax^{2}+by^{2}=1 is

a. xyy_{2}+y_{1}^{2}+yy_{1}=0

b. xyy_{2}+xy_{1}^{2}-yy_{1}=0

c. xyy_{2}-xy_{1}^{2}+yy_{1}=0

d. none of these

Correct option: (b)

The different equation which represents the family of curves y = e^{Cx} is

- y
_{1}= C^{2}y - xy
_{1}- ln y =0 - x ln y = yy
_{1} - y ln y = xy
_{1}

Correct option: (d)

Note: log is considered same as ln.

- u = log x
- u = e
^{z} - u = (log z)
^{-1} - u = (log z)
^{2}

Correct option: (c)

Correct option:(a)

- m = 3, n = 3
- m = 3, n =2
- m = 3, n =5
- m =3, n =1

Correct option: (b)

Correct option: (d)

Correct option: (d)

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

- x =Cy
^{2} - y =Cx
^{2} - x
^{2}=Cy^{2} - y =Cx

Correct option: (a)

The solution of the differential equation x dx +y dy =x^{2}y dy -y^{2} x dx , is

- x
^{2}-1 = C (1+y^{2}) - x
^{2}+1=C (1-y^{2}) - x
^{3}-1=C (1+y^{3}) - x
^{3}+1=C (1-y^{3})

Correct option:(a)

Correct option:

Correct option: (b)

- k =0
- k > 0
- k < 0
- none of these

Correct option:(c)

- tan
^{-1 }x-tan^{-1}y = tan^{-1}C - tan
^{-1 }y-tan^{-1}x = tan^{-1}C - tan
^{-1 }y ± tan^{-1}x = tan C - tan
^{-1 }y +tan^{-1}x = tan^{-1}C

Correct option: (d)

Correct option: (b)

Correct option:(d)

- p < q
- p = q
- p > q
- none of these

Correct option: (c)

Note: Answer not matching with back answer.

- x
- e
^{x} - log x
- log (log x)

Correct option: (c)

- sec x + tan x
- log (sec x+ tan x)
- e
^{sec}^{ x} - sec x

Correct option: (a)

(a) cos x

(b) tan x

(c) sec x

(d) sin x

Correct option: (c)

(a) 3

(b) 2

(c) 1

(d) Not defined

Correct option: (d)

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

Hence, it is not defined.

(a) 2

(b) 1

(c) 0

(d) Not defined

Correct option:(a)

Highest order of the derivative is 2.

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

(a) 0

(b) 2

(c) 3

(d) 4

Correct option: (d)

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

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

Correct option: (d)

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

Correct option: (b)

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

Correct option: (c)

(a) e^{x} +e^{-y} =C

(b) e^{x }+ e^{y}= C

(c) e^{-x} +e^{y} =C

(d) e-^{x} +e^{-y} =C

Correct option: (a)

(a) y = vx

(b) v = yx

(c) x = vy

(d) x = v

Correct option: (c)

Which of the following is a homogeneous differential equation?

(a) (4x+6y+5) dy-(3y +2x+4) dx =0

(b) xy dx -(x^{3}+y^{3})dy =0

(c) (x^{3}+2y^{2})dx+2xy dy =0

(d) y^{2 }dx+(x^{2}-xy-y^{2}) dy =0

Correct option: (d)

- e
^{-x} - e
^{-y} - 1/x
- x

Correct option: (c)

Correct option:(d)

(a) xy =C

(b) x = Cy^{2}

(c) y = Cx

(d) y = Cx^{2}

Correct option:(c)

Correct option: (c)

The general solution of the differential equation e^{x}dy+(ye^{x}+2x) dx =0 is

(a) x e^{y}+x^{2}=C

(b) x e^{y}+y^{2}=C

(c) y e^{x}+x^{2}=C

(d) y e^{y}+x^{2}=C

Correct option: (c)

## Why CBSE Class 12 Science Maths solutions are important?

Maths is a subject which requires practising a variety of problems to understand concepts clearly. By solving as many problems as you can, you’ll be able to train your brain in thinking the logical way to solve maths problems. For practising problems, study materials such as sample papers, previous year papers, and NCERT solutions are needed.

Some of the best Maths experts work with us to give you the best solutions for Maths textbook questions and sample paper questions. Chapter-wise NCERT solutions for Class 12 Science Maths can be easily accessible on TopperLearning. Use these solutions to practise problems based on concepts such as direction ratios, probability, area between lines, inverse trigonometric functions, and more.

To prepare for your Maths exam, you need to attempt solving different kinds of Maths questions. One of the best ways to assess your problem-solving abilities is to attempt solving previous year papers with a set timer. Our Maths solutions will come in handy to help you with checking your answers and thus, improving your learning experience. So, to score more marks in your Class 12 board exams, use our Maths solutions that will enable you with the appropriate preparation.

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