# NCERT Solutions for Miscellaneous Exercise Chapter 9 Class 12 - Differential Equations

At the end of all the chapters of NCERT Class 12 Maths book, there is an exercise known as miscellaneous exercise. This covers questions from the whole chapter. NCERT solutions for Class 12 Maths chapter 9 miscellaneous exercise take a tour through all the concepts covered in this NCERT chapter. Class 12 Maths chapter 9 miscellaneous exercise solutions are a bit more lengthy compared to other exercises of differential equations. Class 12 Maths chapter 9 miscellaneous solutions covers the complete chapter through questions. If one student is able to solve miscellaneous exercise chapter 9 Class 12 without looking to the solution, then it implies he has understood the concepts covered in the chapter. The ideas covered in the last 6 exercises are used in the NCERT solutions for Class 12 Maths chapter 9 miscellaneous exercise. Before solving miscellaneous exercise chapter 9 Class 12 it’s better to cover the following exercises.

Differential Equations exercise 9.1

Differential Equations exercise 9.2

Differential Equations exercise 9.3

Differential Equations exercise 9.4

Differential Equations exercise 9.5

Differential Equations exercise 9.6

## **Differential Equations Class 12 Chapter 9 ****-Miscellaneous Exercise**** **

** Question:1 ** Indicate Order and Degree.

(i)

** Answer: **

Given function is

We can rewrite it as

Now, it is clear from the above that, the highest order derivative present in differential equation is

Therefore, the order of the given differential equation ** is 2 **

Now, the given differential equation is a polynomial equation in its derivative y '' and y 'and power raised to y '' is 1

Therefore, it's degree is 1

** Question:1 ** Indicate Order and Degree.

(ii)

** Answer: **

Given function is

We can rewrite it as

Now, it is clear from the above that, the highest order derivative present in differential equation is y'

Therefore, ** order ** of given differential equation ** ** ** is 1 **

Now, the given differential equation is a polynomial equation in it's dervatives y 'and power raised to y ' is 3

Therefore, it's ** degree is 3 **

** Question:1 ** Indicate Order and Degree.

(iii)

** Answer: **

Given function is

We can rewrite it as

Now, it is clear from the above that, the highest order derivative present in differential equation is y''''

Therefore, order of given differential equation ** ** is 4

Now, the given differential equation is not a polynomial equation in it's dervatives

Therefore, it's degree is not defined

** Question:2 ** Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

** Answer: **

Given,

Now, differentiating both sides w.r.t. x,

Again, differentiating both sides w.r.t. x,

Therefore, the given function is the solution of the corresponding differential equation.

** Question:2 ** Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(ii)

** Answer: **

Given,

Now, differentiating both sides w.r.t. x,

Again, differentiating both sides w.r.t. x,

Therefore, the given function is the solution of the corresponding differential equation.

** Question:2 ** Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(iii)

** Answer: **

Given,

Now, differentiating both sides w.r.t. x,

Again, differentiating both sides w.r.t. x,

Therefore, the given function is the solution of the corresponding differential equation.

** Question:2 ** Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(iv)

** Answer: **

Given,

Now, differentiating both sides w.r.t. x,

Putting values in LHS

Therefore, the given function is the solution of the corresponding differential equation.

** Question:3 ** Form the differential equation representing the family of curves given by , where * a * is an arbitrary constant.

** Answer: **

Given equation is

we can rewrite it as

-(i)

Differentiate both the sides w.r.t x

-(ii)

Put value from equation (ii) in (i)

Therefore, the required differential equation is

** Question:4 ** Prove that is the general solution of differential equation , where c is a parameter.

** Answer: **

Given,

Now, let y = vx

Substituting the values of y and y' in the equation,

Integrating both sides we get,

Now,

Let

Now,

Let v ^{ 2 } = p

Now, substituting the values of I _{ 1 } and I _{ 2 } in the above equation, we get,

Thus,

** Question:5 ** Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

** Answer: **

Now, equation of the circle with center at (x,y) and radius r is

Since, it touch the coordinate axes in first quadrant

Therefore, x = y = r

-(i)

Differentiate it w.r.t x

we will get

-(ii)

Put value from equation (ii) in equation (i)

Therefore, the differential equation of the family of circles in the first quadrant which touches the coordinate axes is

** Question:6 ** Find the general solution of the differential equation

** Answer: **

Given equation is

we can rewrite it as

Now, integrate on both the sides

Therefore, the general solution of the differential equation is

** Question:7 ** Show that the general solution of the differential equation is given by , where * A * is parameter.

** Answer: **

Given,

Integrating both sides,

Let

Let A = ,

Hence proved.

** Question:8 ** Find the equation of the curve passing through the point whose differential equation is

** Answer: **

Given equation is

we can rewrite it as

Integrate both the sides

Now by using boundary conditiond, we will find the value of C

It is given that the curve passing through the point

So,

Now,

Therefore, the equation of the curve passing through the point whose differential equation is is

** Question:9 ** Find the particular solution of the differential equation , given that when .

** Answer: **

Given equation is

we can rewrite it as

Now, integrate both the sides

Put

Put again

Put this in our equation

Now, by using boundary conditions we will find the value of C

It is given that

y = 1 when x = 0

Now, put the value of C

Therefore, the particular solution of the differential equation is

** Question:10 ** Solve the differential equation

** Answer: **

Given,

Let

Differentiating it w.r.t. y, we get,

Thus from these two equations,we get,

** Question:11 ** Find a particular solution of the differential equation , given that , when . (Hint: put )

** Answer: **

Given equation is

Now, integrate both the sides

Put

Now, given equation become

Now, integrate both the sides

Put again

Now, by using boundary conditions we will find the value of C

It is given that

y = -1 when x = 0

Now, put the value of C

Therefore, the particular solution of the differential equation is

** Question:12 ** Solve the differential equation .

** Answer: **

Given,

This is equation is in the form of

p = and Q =

Now, I.F. =

We know that the solution of the given differential equation is:

** Question:13 ** Find a particular solution of the differential equation , given that .

** Answer: **

Given equation is

This is type where and

Now,

Now, the solution of given differential equation is given by relation

Now, by using boundary conditions we will find the value of C

It is given that y = 0 when

at

Now, put the value of C

Therefore, the particular solution is

** Question:14 ** Find a particular solution of the differential equation , given that when

** Answer: **

Given equation is

we can rewrite it as

Integrate both the sides

Put

put again

Put this in our equation

Now, by using boundary conditions we will find the value of C

It is given that y = 0 when x = 0

at x = 0

Now, put the value of C

Therefore, the particular solution is

** Question:15 ** The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

** Answer: **

Let n be the population of the village at any time t.

According to question,

Now, at t=0, n = 20000 (Year 1999)

Again, at t=5, n= 25000 (Year 2004)

Using these values, at t =10 (Year 2009)

Therefore, the population of the village in 2009 will be 31250.

** Question:16 ** The general solution of the differential equation is

(A)

(B)

(C)

(D)

** Answer: **

Given equation is

we can rewrite it as

Integrate both the sides

we will get

Therefore, answer is (C)

** Question:17 ** The general solution of a differential equation of the type is

(A)

(B)

(C)

(D)

** Answer: **

Given equation is

and we know that the general equation of such type of differential equation is

Therefore, the correct answer is (C)

** Question:18 ** The general solution of the differential equation is

(A)

(B)

(C)

(D)

** Answer: **

Given equation is

we can rewrite it as

It is type of equation where

Now,

Now, the general solution is

Therefore, (C) is the correct answer

**More About NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise**

Eighteen questions for practising the whole chapter are present in the miscellaneous exercise chapter 9 Class 12. All these questions are solved in Class 12 Maths chapter 9 miscellaneous exercise solutions. The Class 12 Maths chapter 9 miscellaneous solutions are given a stepwise manner and can be accessed for free. Students can also download the NCERT solutions for Class 12 Maths chapter 9 miscellaneous exercise using webpage download options available online.

**Also Read| **Differential Equations Class 12th Notes

**Benefits of NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise**

By going through the Class 12 Maths chapter 9 miscellaneous exercise solutions students can have better exposure to the concepts detailed in the chapter.

- All the questions in the miscellaneous exercise chapter 9 Class 12 are important and are important from an exam point of view.

**Also see-**

- NCERT Exemplar Solutions Class 12 Maths Chapter 9
NCERT Solutions for Class 12 Maths Chapter 9

**NCERT Solutions Subject Wise**

- NCERT Solutions Class 12 Chemistry
- NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Biology

NCERT Solutions for Class 12 Mathematics

**Subject Wise NCERT Exemplar Solutions**

- NCERT Exemplar Class 12 Maths
- NCERT Exemplar Class 12 Physics
NCERT Exemplar Class 12 Chemistry

NCERT Exemplar Class 12 Biology