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RD Sharma Solutions for Class 12 are made to help students perform well and learn various concepts. RD Sharma books have a good reputation for being informative and comprehensive. Class 12 Maths is a challenging subject that requires a lot of practice. RD Sharma Class 12th Exercise RE material helps students practice better as it contains solved examples that have step-by-step answers. This is an excellent alternative for preparation as it follows the CBSE syllabus and is prepared by subject experts. RD Sharma Class 12 Chapter 21 Exercise RE also helps students with Revision as the solutions are easy to understand and are available in one place.

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Differential Equations Exercise Revision Exercise Question 1(i)

Order, degree

Check the highest order of derivate to find the order and check the power of derivative to find degree.

Highest order derivative

Order

Degree= Power of

Degree** **

Differential Equations Exercise Revision Exercise Question 1(ii)

Order,Degree

Check the highest order of derivate to find the order and check the power of highest derivative to find degree.

Order

Degree Power of

Degree

Differential Equations Exercise Revision Exercise Question 1(iii)

Degree ,Order

Check the highest order of derivate to find the order and check the power of highest derivative to find degree.

Highest order derivative

Order

Degreepower of

Degree

Differential Equations Exercise Revision Exercise Question 1(iv)

Check the highest order of derivate to find the order and check the power of highest derivative to find degree.

Highest order derivative

Order

Degreepower of

Degree

Differential Equations Exercise Revision Exercise Question 1(v)

Check the highest order of derivate to find the order and check the power of highest derivative to find degree.**Given:****Solution:**

Highest order derivative

Order

Degreepower of

Degree

Differential Equations Exercise Revision Exercise Question 1(vii)

Degree is not defined, order

Check the highest order of derivate to find the order and check the power of highest derivative to find degree.

Highest order derivative

Order

As the equation cannot be expressed as a polynomial of derivative

Degree is not defined

Differential Equations Exercise Revision Exercise Question 2

Verified

Find double derivatives of given equation and put values to verify

Prove that is the solution of

Put it in equation

Hence proved.

Differential Equations Exercise Revision Exercise Question 3(i)

Verified

Find the first and second derivative of given function and put values in differential equation to verify

Solution:

Put in given differential equation,

Hence verified

Differential Equations Exercise Revision Exercise Question 3(ii)

Answer:

Verified

Hint:

Find the first and second derivative of given function and put in differential equation to verify

Given:

Solution:

Put in differential equation,

Hence verified

Differential Equations Exercise Revision Exercise Question 3(iii)

Answer:Verified

Hint:

Find the first and second derivative of given function and put in differential equation to verify

Given:

Solution:

Put in differential equation,

Hence verified

Differential Equations Exercise Revision Exercise Question 3(iv)

Verified

Find the first derivative of given function and put value in differential equation

Put in differential equation,

LHS

Multiply and divide by

Where,

Therefore, LHS RHS..Hence verified.

Differential Equations Exercise Revision Exercise Question 3(v)

Verified

You must firstly solve the first derivative and put value in differential equation

Put in differential equation

LHSRHS

Hence verified

Differential Equations Exercise Revision Exercise Question 3(vi)

Verified

Find first derivative and put value in differential equation to verify

Put in differential equation

Hence verified

Differential Equations Exercise Revision Exercise Question 4

You must have the knowledge about the curves and make differential equation.

Where, is an arbitrary constant

Substitute in

which is the required equation

Differential Equations Exercise Revision Exercise Question 5

The number of constants is equal to the number of time we differentiate

, where are constants

Here, there are two constants. So, we differentiate twice

Again,

which is the required differential equation.

Differential Equations Exercise Revision Exercise Question 6

You must know the equation of parabola

Family of parabola having vertex at origin and axis along positive direction of x-axis

Since parabola has axis along positive x-axis , its equation is

… (i)

Differentiate,

Putting value in (i)

which is the required differential equation

Differential Equations Exercise Revision Exercise Question 7

You must know about the equation of circle

Family of circles having centre on y-axis and radius 3 units

General equation of circle is

Given centre is on y-axis

Centre and Radius

Hence, our equation is

Differentiate with respect to ,

Put value in (i)

Differential Equations Exercise Revision Exercise Question 8

Answer:Hint:

You must know about the parabolas to form an equation

Given:

Family of parabola having vertex at origin and axis along the positive y-direction

Solution:

… (i)

Where, is constant parameter

Differentiate with respect to ,

Differential Equations Exercise Revision Exercise Question 9

Answer:Hint:

You must know about the equation of ellipses

Given:

Family of ellipses having foci on y-axis and centre at origin

Solution:

Ellipse whose foci on y-axis

[Two constants, differentiate twice]

Again differentiate,

Differential Equations Exercise Revision Exercise Question 10

[Two constants, differentiate twice]

Again differentiate,

Differential Equations Exercise Revision Exercise Question 11

Hence proved

Differential Equations Exercise Revision Exercise Question 12

Answer: Hence verifiedHint: Find first and second derivative of given equation and put in differential equation to be verified.

Given:

Solution:

…differentiating w.r.t to x

..differentiating agin w.r.t to

Now, differential equation is

Hence verified

Differential Equations Exercise Revision Exercise Question 13

Answer: is the solution of the given differential equationHint: Find first and second order derivative and put values in differential equation to be verified.

Given:

Solution:

Thus is the solution of the given differential equation

Differential Equations Exercise Revision Exercise Question 14

…Differentiating w.r.t to x,

Now, differential eq.

Hence, Proved.

Differential Equations Exercise Revision Exercise Question 15

Differential Equations Exercise Revision Exercise Question 16

..where n,k are constants

Differentiating w.r.t to ,

Differentiating again,

Again Differentiating,

Differential Equations Exercise Revision Exercise Question 17

Differentiate w.r.t x,

Again, differentiate w.r.t x

..taking 2 common

We have,

Again differentiate,

On substituting values of (b),

Differential Equations Exercise Revision Exercise Question 18

Now,

Let

Let

Differential Equations Exercise Revision Exercise Question 19

Hint: Use the formula of

Given:

Solution:

Integrating both sides, we get

Differential Equations Exercise Revision Exercise Question 20

Hint: Separate y & x and than integrate both sides

Given:

Solution:

Integrating both sides, we get

Differential Equations Exercise Revision Exercise Question 21

integrate both sides

Differential Equations Exercise Revision Exercise Question 22

Hint: You must know about the formula of

Given:

Solution:

Differential Equations Exercise Revision Exercise Question 23

Differential Equations Exercise Revision Exercise Question 24

Hint: Use substitution method

Given:

Solution:

Integrating both sides, we get,

Let,

Differentiate w.r.t x

Differential Equations Exercise Revision Exercise Question 25

Hint: Apply integration by parts method and formula of

Given:

Solution:

(integrate both sides)

Let

Now,

Differential Equations Exercise Revision Exercise Question 26

Hint: You must know about trigonometric identities.

Given:

Solution:

Integrating both sides

Differential Equations Exercise Revision Exercise Question 27

Differential Equations Exercise Revision Exercise Question 28

Hint: Use the formula of

Given:

Solution:

Differential Equations Exercise Revision Exercise Question 29

Hint: Apply substitution method.

Given:

Solution:

Let

Differential Equations Exercise Revision Exercise Question 30

(integrating both sides)

Let log y=t

Differential Equations Exercise Revision Exercise Question 31

Integrating both sides, we get,

Substituting

Again substituting,

Substituting values of A and B in , we get,

Now considering R.H.S of (II), we have,

Here, putting and differentiate both sides, we get,

Now substituting the values of and in (I)

Differential Equations Exercise Revision Exercise Question 32

integrating both sides,

Integrating by parts,

Differential Equations Exercise Revision Exercise Question 33

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

Integrating both sides,

Differential Equations Exercise Revision Exercise Question 34

Answer:\Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:..........(I)

Put x + y = t and differentiate both sides. We get,

Compare with equation (I),

Now, integrating both sides,

Differential Equations Exercise Revision Exercise Question 35

Let

Now, integrating both sides,

Hence,

Differential Equations Exercise Revision Exercise Question 36

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

Let and differentiating both sides,

Now, integrating both sides,

Put the value of u

Differential Equations Exercise Revision Exercise Question 37

Put y = v x and differentiate both sides w.r.t x

So, equation (I) becomes

Now, integrating both sides

Differential Equations Exercise Revision Exercise Question 38

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

.....................(1)

Put , y=vx and differentiate both side.

Eq. (1) becomes,

Now, integrating both sides,

Put value of

(where A is integration constant)

Differential Equations Exercise Revision Exercise Question 39

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

Putting , we get,

Integration both sides we get

Differential Equations Exercise Revision Exercise Question 40

The above equation is in form of

Where p = -cot x and q = cosec x

Integrating factor =

Considering

Now, General solution is,

Differential Equations Exercise Revision Exercise Question 41

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

The above equation look like,

Where p = -tan x and q = -2 sin x

Integrating factor =

we have

Hence, now the solution of differential equation is,

Differential Equations Exercise Revision Exercise Question 42

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

Comparing with,

So, the solution is,

Differential Equations Exercise Revision Exercise Question 43

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

Compare with , we get,

Now the solution is,

Differential Equations Exercise Revision Exercise Question 44

Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution:

Comparing with , we get,

The solution is,

Differential Equation Exercise Revision Exercise (RE) Question 45

Answer :Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution :

Comparing with , we get,

The solution is,

Put differentiating both side,

Differential Equation Exercise Revision Exercise (RE) Question 46

Answer :Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution :

Now, Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 47

Answer :Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution :

split

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 48

Answer :Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution :

Put , differentiate both sides,

Integrating both sides,

put,

Put value of t

put value of

Differential Equation Exercise Revision Exercise (RE) Question 49

Answer :Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution :

putting and differentiate,

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 50

Answer :Hint: you must know the rules of solving differential equation and integrations.

Given:

Solution :

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 51

Answer :Hint : You must know the rules of solving differential equation and integration.

Given :

Solution :

Compare with,

where, and

Therefore,

The solution is,

......(i)

Where, ......(ii)

Apply integrating by parts,

.....(iii)

From (i) and (iii), we get,

Is required solution.

Differential Equation Exercise Revision Exercise (RE) Question 52

Answer :Hints : You must know the rules of solving differential equation and integration.

Given :

Solution : .....(i)

Compare with,

Where,

Therefore,

Hence, the solution is ,

Integrating by parts,

is required solution

Differential Equation Exercise Revision Exercise (RE) Question 53

Answer :Hint : You must know the rules of solving differential equation and integration.

Given :

Solution :

Compare with,

When

Therefore,

The solution is ,

....(i)

Where,

Therefore, required solution is

Differential Equation Exercise Revision Exercise (RE) Question 54

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Putting and differentiate

Therefore,

Therefore,

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 55

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Comparing with ,

Where,

Now,

So, the solution is

Now,

Integrating by parts,

Therefore,

Therefore,

Hence, the required solution is

Differential Equation Exercise Revision Exercise (RE) Question 56

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Comparing with,

Where,

Therefore,

So the solution is,

Differential Equation Exercise Revision Exercise (RE) Question 57

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Comparing with,

Now,

So, the solution is

Differential Equation Exercise Revision Exercise (RE) Question 58

Answer :Hint : you must know the rules of solving differential equation and integration

Given :

Solution :

Comparing with,

Where ,

Now,

therefore the solution is

Putting and differentiating both sides

Applying integration both side,

Therefore,

Hence, required solution is ,

Differential Equation Exercise Revision Exercise (RE) Question 59

Answer :Hint : you must know the rules of solving differential equation and integration

Given : ,

Solution :

Comparing with, , we get

Where,

Now,

So, the solution is,

Therefore,

Putiing value of C,

Differential Equation Exercise Revision Exercise (RE) Question 60

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Comparing with,

So, the solution is ,

Now,

Integrating by parts,

By putting value of I

We get the required solution,

Differential Equation Exercise Revision Exercise (RE) Question 61

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Comparing with,

Now,

So the solution is ,

Differential Equation Exercise Revision Exercise (RE) Question 62

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 63

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides ,

Now,

Therefore,

Put value of c,

Differential Equation Exercise Revision Exercise (RE) Question 64 (i)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 64 (ii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 64 (iii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Differential Equation Exercise Revision Exercise (RE) Question 64 (iv)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Put and differentiating,

Therefore,

Differential Equation Exercise Revision Exercise (RE) Question 64 (v)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides

Integrating by parts,

Put

Therefore,

Differential Equation Exercise Revision Exercise (RE) Question 64 (vi)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides

Differential Equation Exercise Revision Exercise (RE) Question 65 (i)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Let,

Comparing both sides,

Therefore,

Therefore,

Now,

Given,

Therefore,

Therefore the solution is ,

Differential Equation Exercise Revision Exercise (RE) Question 65 (ii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Now,

Therefore,

Hence,

Differential Equation Exercise Revision Exercise (RE) Question 65 (iii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Integrating both sides,

Now,

Therefore,

Put value of c ,

Differential Equation Exercise Revision Exercise (RE) Question 66 (i)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

.....(i)

STEP : 3

So, our equation become

....(ii)

Solving ,

Diff.w.r.t.v

Put value of v and dv in

Solving

From (ii)

Replace v by

Multiply both side by 2

Differential Equation Exercise Revision Exercise (RE) Question 66 (ii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution : given differential equation

It is a homogeneous differential equation

Put

Eqn (i) becomes

Integrating both sides

Differential Equation Exercise Revision Exercise (RE) Question 66 (iii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Step : 2

Putting

Step : 3

Solve by

Put

Diff w.r.t.x

Put value of

Integrate both sides

Put

So our equation

Put value of t

Differential Equation Exercise Revision Exercise (RE) Question 66 (iv)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution : differential equation is in form of

Solution is

Integrate by

From (i)

Differential Equation Exercise Revision Exercise (RE) Question 66 (v)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution :

Divide both sides by x

Differentiate equation in the form ,

Where,

Solution of differential equation is

Differential Equation Exercise Revision Exercise (RE) Question 66 (vi)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution : put in form

...(i)

Step : find P and Q

Compare (i) with

Find integration factor I.F

Step : 4

Putting value

Solving I

Substituting I in eq (ii)

Differential Equation Exercise Revision Exercise (RE) Question 66 (vii)

Answer :Hint : You must know the rules of solving differential equation and integration

Given :

Solution : put in form

Step : 2

Find P and Q by comparing ,

Step : 3

Find integrating factor

Step : 4

Solution of equation

Putting values

divide by

Differential Equation Exercise Revision Exercise (RE) Question 66 (viii)

Answer :Hint : : integrate by applying integration of

Given :

Solution :

Differential equation is in the form of

,

Putting I.F

Solution is

Differential Equation Exercise Revision Exercise (RE) Question 66 (ix)

Answer :Hint : integrate by applying integration of sec x and tan x

Given :

Solution : differential equation is of the form

Finding integrating factor

Solution is

Differential Equation Exercise Revision Exercise (RE) Question 66 (x)

Answer :Hint : integrate by applying integration of and

Given :

Solution : convert the given differential equation is of the form

Divide both sides by x

Differentiate

Solution is

Differential Equation Exercise Revision Exercise (RE) Question 66 (xi)

Answer :Hint : you integrate by integrating

Given :

Solution :

Divide bot side by

...(i)

By comparing (i) with

....(ii)

Now enq (ii) becomes

Differential Equation Exercise Revision Exercise (RE) Question 66 (xii)

Answer :Hint : integrate by applying integration of

Given :

Solution : given equation

Divide both sides by dx

Divide by

Comparing above equation with

Differential Equation Exercise Revision Exercise (RE) Question 66 (xiii)

Answer :Hint : you integrate by applying integration of

Given :

Solution : Put in the form of

Divide by

.....(i)

Find and

Comparing (i)

Find I.F

Putting value ......(ii)

Put value of I in (ii)

Divide by

Differential Equation Exercise Revision Exercise (RE) Question 66 (xiv)

Answer :Hint : integrate by applying integration of

Given :

Solution :

This is not in the form of

.....(i)

Find P and Q Where

Find I.F

Solution will be

Differential Equation Exercise Revision Exercise (RE) Question 66 (xv)

Answer :Hint : integrate by applying integration of

Given :

Solution :

This is not in the foem of

Where

Step : 3 find integration factor

Step : 4

Solution is

Differential Equation Exercise Revision Exercise (RE) Question 67 (i)

Answer :Hint : Use the formula

Given :

Solution :

Divide by

This is a linear differential equation of the form

Here,

The integrating factor of this differential equation is

We have,

Therefore,

Hence the solution of the differential equation is

...(i)

We know ,

Therefore, ....(ii)

Now when

Therefore,

By (ii)

Differential Equation Exercise Revision Exercise (RE) Question 67 (ii)

Answer :Hint : using variable separable method and substituting the values

Given :

Solution :

Now let ,

Therefore, is a homogeneous function of degree 0 .

Putting

Diff w.r.t.x

Integrating both sides

Putting in integral

Now again putting back the value of

....(ii)

Now, when

Put in (ii)

Differential Equation Exercise Revision Exercise (RE) Question 67 (iii)

Answer :Hint : using variable separable method and substituting the values

Given :

Solution :

The differential equation can be written as

......(i)

Let

Finding

Therefore , is a homogenous function of degree zero.

Putting

Diff w.r.t.x

Putting value of

Integrating both sides we get ,

Using

Now putting value of v i.e , y/x

Squaring both sides

Putting back in (ii)

Differential Equation Exercise Revision Exercise (RE) Question 68

Answer :Hint : Using variable separable method

Given :

Solution :

Integrating both sides

....(i)

.....(ii)

Since the curve passes through (1,1)

Putting in (ii)

Put in (ii)

i.e ,

Hence the equation of curve is

Differential Equation Exercise Revision Exercise (RE) Question 69

Answer :Hint : use variable separable method

Given : slope of the tangent to the curve at any

Solution : Slope of tangent

Integrating both sides

Given that equation passes through (-2, 3)

Putting in (i)

Putting in (i)

is the perticular solution of the equation

Differential Equation Exercise Revision Exercise (RE) Question 70

Answer :Hint : using integration by parts

Given :

Solution :

Integrating both sides

....(i)

Using integration by parts

Let

Put in (i), we get,

....(ii)

Given curve passes through (0,0)

Putting x = 0 , y = 0 in equation

Therefore ,

Putting value of C in (ii)

Differential Equation Exercise Revision Exercise (RE) Question 71

Answer:Given:

At any point of a curve , the slope of the tangent is twice the slope of the line segment joining the point of contact to the point

Hint:

Using variable separable method and substituting the values.

Explanation:

Slope of the tangent to the curve

Slope of line segment joining and

Given at point . Slope of tangent is twice of line segment

Integrating both sides.

Since curve passes through (-2,1)

Put in (1)

Put value of c in eq (1)

Hence the equation of the curve is

Differential Equation Exercise Revision Exercise (RE) Question 72

Answer :Given :

Hint:

Using variable separable method.

Explanation:

We know that the slope of the tangent at of a Curve is

Given slope of tangent at is

So is homogeneous function of degree 0

Given equation is homogeneous differential equation

Now Put in (1)

Differentiate with respect to x

Integrating both sides,

Solving

Put

Differentiate with respect to v

Putting back

By (2)

Putting

Remove log of both sides.

Hence Proved

Differential Equation Exercise Revision Exercise (RE) Question 73

Answer:Given: The slope of the tangent to the curve at any point is equal to the sum of x coordinate and the product of x and y coordinate of that point

Hint: You must know about integrating factor

Explanation: Slope of the tangent to the curve at

Given that,

Slope of the tangent to the curve at point is equal to the sum of x coordinate and the product of x and y coordinate of that point

So our equation becomes,

Differential equation is of the form

Where

If

Solution is

Putting ,

Thus our equation becomes

Since curve passes through (0,1)

Putting

Putting value of c in (1)

Equation of the curve is

Differential Equation Exercise Revision Exercise (RE) Question 74

Answer :Given: The slope of the tangent to the curve at any point is equal to the sum of x coordinate and the product of x and y coordinate of that point

Hint: You must know about integrating factor

Explanation: We know that

Slope of the tangent to the curve at

According to the given

This is of the form

Where

Finding integrating factor

If,

Solution is

Using integration by parts,

Put in (1)

Divide by

Since curve passes through origin,

Putting x=0 and y=0 in (2)

Put vaue of c in (2)

Differential Equation Exercise Revision Exercise (RE) Question 75

Answer:Given: Sum of the coordinate of any point of curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5

Hint: Using integration by parts and integration factor

Explanation: We know that

Slope of the tangent to the curve at

According to the given

So we will take both (+) ve and (-) ve sign and then solve it

Taking (+) ve sign:

Equation is of the form

Where,

Solution is

Divided by

Since the curve passes through -the point (0,2)

Put x=0 and

Equation of the curve is

Taking (-) ve sign:

Equation is of the form

Where,

Solution is

[Using integration by parts]

Divided by

Since the curve passes through the point (0,2)

Put x=0 and

Put in (2) we get,

Equation of the curve is

Differential Equation Exercise Revision Exercise (RE) Question 76

Answer :Given: The slope of the tangent to the curve at any point is the reciprocal of twice the ordinate at that point. Also the curve passes through the point(4,3)

Hint: Using variable separable method

Explanation: Let be any point on the curve

Slop of the tangent at

Acc to given condition,

Integrating both sides,

Since the curve passes through (4,3) then,

Hence the required equation of the curve is

Differential Equation Exercise Revision Exercise (RE) Question 77

Answer :Given: The decay rate of radium at any time t is proportional to its mass at that time.

Hint: Using variable separable method

Explanation: Acc to given,

Integrating both sides

Put in (1)

We have to find the time when

Put in (2)

Differential Equation Exercise Revision Exercise (RE) Question 78

Answer: 0.04%Given: Radius disintegrates at a rate proportional to the amount of radium present at the moment

Hint: Using variable separable method

Explanation: Let A be the amount radium at that time

According to given,

Integrating both sides,

Let be the initial amount at

Now,

Differential Equation Exercise Revision Exercise (RE) Question 79

Answer :Given: A wet porous substance in the wet air loses its moisture at a rate proportional to the moisture content

Hint: Using variable separable method

Explanation: Let M be the moisture content at any time A

According to given,

when k be any constant and [-ve sign. ? of it losses its moisture]

Integrating both the sides.

Put in (1)

Given tht at

Put in (2)

Put in (2)

Let at

moisture = 95%

Remaining = 5 %

Put in (3)

Class 12 maths book consists of eleven exercises from the 21st chapter. There are 113 revision-based exercises in this material. The concepts on which these questions are framed are, differentiating various sums, integration factor of differential equations, formation of differential equations, solving first-order differential equations, and many more. It may seem to be quite enormous, but all it takes is a good practice with the RD Sharma Class 12th Exercise solution book. Therefore, the importance of the RD Sharma Class 12 Solutions Differential Equations Ex RE book is ineffable.

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RD Sharma Class 12th Exercise RE is available on Career360’s website for free. Students can access this material by searching the title name on the website.

As RD Sharma Class 12 Solutions Chapter 21 Ex RE follows the CBSE syllabus, students can use it for revision as well as homework completion. As Maths is a vast subject, studying from this material can save a lot of time and help students get a good hold on the subject.

RD Sharma Class 12th Exercise RE is an ideal solution for exam preparation. It acts as a guide for students to help them score good marks in exams.

**RD Sharma Chapter wise Solutions**

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. What is the most used solution book for class 12 students to prepare for the maths exam?

The students can use the Class 12 RD Sharma Chapter 21 Exercise RE Solution material to prepare well for the public exams. It helps the students to understand the mathematical concepts effortlessly.

2. What is the cost of the RD Sharma solution books to download from the Career360 website?

The students need not pay even a single penny for the RD Sharma class 12th exercise book and the other RD Sharma books when downloaded from the Career360 website.

3. Can the students trust the RD Sharma solution material for the public exam preparation?

The students must practice every day to learn the concepts in-depth. For this, they can use the RD Sharma reference books to get ready for the board exams.

4. Which guide can the class 12 students use to learn the Differential Equation concept?

The differential equation chapter is a bit challenging concept for the 12th graders. This can be cleared by using the Class 12 RD Sharma Chapter 21 Exercise RE Solution reference book.

5. How many exercises are available in Chapter 21 of the class 12 maths book?

There are eleven exercises in total, ex 21.1 to ex 21.11. The students can easily follow the syllabus without any confusion using the RD Sharma solution books.

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