9- DIFFERENTIAL EQUATIONS


Here we are going through the ncert solutions for class 12 maths chapter 9 – differential equations so before going through the ncert solutions make sure to go through the textbook that helps you to understand the solutions more easily.

 ​​​​ Differential Equations

 

Exercise 9.1

Question 1:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest​​ order derivative present in the differential equation is. Therefore, its order is four.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

 

Question 2:

Determine order and degree(if​​ defined) of differential equation 

ANSWER:

The given differential equation is:

The highest order derivative present in the differential equation is. Therefore, its order is one.

It is a polynomial equation in. The highest power raised tois 1.​​ Hence, its degree is one.

 

Question 3:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order derivative present in the given differential equation is. Therefore, its order is two.

It is a polynomial equation inand. The power raised tois 1.

Hence, its degree is one.

 

Question 4:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order derivative present in the given differential equation is. Therefore, its order is 2.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

 

Question 5:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order derivative present in the​​ differential equation is. Therefore, its order is two.

It is a polynomial equation inand the power raised tois 1.

Hence, its degree is one.

 

Question 6:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order​​ derivative present in the differential equation is. Therefore, its order is three.

The given differential equation is a polynomial equation in.

The highest power raised tois 2. Hence, its degree is 2.

 

Question 7:

Determine order and degree(if defined)​​ of differential equation 

ANSWER:

The highest order derivative present in the differential equation is. Therefore, its order is three.

It is a polynomial equation in. The highest power raised tois 1. Hence, its degree is 1.



Question 8:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order derivative present in the differential equation is. Therefore, its order is one.

The given differential equation is a polynomial equation inand the highest power raised tois one. Hence, its degree is one.

 

Question 9:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order derivative present in the differential equation is. Therefore, its order is two.

The given​​ differential equation is a polynomial equation inandand the highest power raised tois one.

Hence, its degree is one.

 

Question 10:

Determine order and degree(if defined) of differential equation 

ANSWER:

The highest order derivative present in the​​ differential equation is. Therefore, its order is two.

This is a polynomial equation inandand the highest power raised tois one. Hence, its degree is one.

 

Question 11:

The degree of the differential equation

is

(A) 3 (B) 2 (C) 1 (D) not​​ defined

ANSWER:

The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.

Hence, the correct answer is D.

 

Question 12:

The order of the differential equation

is

(A) 2 (B) 1 (C) 0 (D) not​​ defined

ANSWER:

The highest order derivative present in the given differential equation is. Therefore, its order is two.

Hence, the correct answer is A.



EXERCISE 9.2

Question 1:

ANSWER:

Differentiating both sides of this equation with​​ respect to x, we get:

Now, differentiating equation (1) with respect to x, we get:

Substituting the values ofin the given differential equation, we get the L.H.S. as:

Thus, the given function is the solution of the corresponding​​ differential equation.

 

Question 2:

ANSWER:

Differentiating both sides of this equation with respect to x, we get:

Substituting the value ofin the given differential equation, we get:

L.H.S. == R.H.S.

Hence, the given function is the​​ solution of the corresponding differential equation.

 

Question 3:

ANSWER:

Differentiating both sides of this equation with respect to x, we get:

Substituting the value ofin the given differential equation, we get:

L.H.S. == R.H.S.

Hence, the​​ given function is the solution of the corresponding differential equation.

 

Question 4:

ANSWER:

Differentiating both sides of the equation with respect to x, we get:

L.H.S. = R.H.S.

Hence, the given function is the solution of the​​ corresponding differential equation.

 

Question 5:

ANSWER:

Differentiating both sides with respect to x, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of the​​ corresponding differential equation.

 

Question 6:

ANSWER:

Differentiating both sides of this equation with respect to x, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of​​ the corresponding differential equation.

 

Question 7:

ANSWER:

Differentiating both sides of this equation with respect to x, we get:

 L.H.S. = R.H.S.

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

 

Question 8:

ANSWER:

Differentiating both sides of the equation with respect to x, we get:

Substituting the value ofin equation (1), we get:

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

 

Question 9:

ANSWER:

Differentiating both sides of this equation with respect to x, we get:

Substituting the value ofin the given differential equation, we get:

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

 

Question 10:

ANSWER:

Differentiating both sides of this equation with respect to x, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of the corresponding​​ differential equation.

 

Question 11:

The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0 (B) 2 (C) 3 (D) 4

ANSWER:

We know that the number of constants in the general solution of a​​ differential equation of order n is equal to its order.

Therefore, the number of constants in the general equation of fourth order differential equation is four.

Hence, the correct answer is D.

 

Question 12:

The numbers of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3 (B) 2 (C) 1 (D) 0

ANSWER:

In a particular solution of a differential equation, there are no arbitrary constants.

Hence, the correct answer is D.



EXERCISE 9.3

Question 1:

ANSWER:

Differentiating both sides of the given equation with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Hence, the required differential equation of the given curve is

 

Question 2:

ANSWER:

Differentiating both sides with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Dividing equation (2) by equation (1), we get:

This is the required differential equation of the given curve.

 

Question 3:

ANSWER:

Differentiating both sides with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Multiplying equation (1) with (2) and then adding it to equation (2), we get:

Now, multiplying equation (1) with 3 and​​ subtracting equation (2) from it, we get:

Substituting the values of in equation (3), we get:

This is the required differential equation of the given curve.

 

Question 4:

ANSWER:

Differentiating both sides with respect to x, we get:

Multiplying equation (1) with 2 and then subtracting it from equation (2), we get:

y'−2y=e2x(2a+2bx+b)−e2x(2a+2bx)⇒y'−2y=be2x                                      ...(3)y'-2y=e2x2a+2bx+b-e2x2a+2bxy'-2y=be2x                                      ...(3)

Differentiating both sides with respect to x, we get:

y''−2y'=2be2x                        ...(4)y''-2y'=2be2x                        ...4

Dividing equation (4) by equation (3), we get:

This is​​ the required differential equation of the given curve.

 

Question 5:

ANSWER:

Differentiating both sides with respect to x, we get:

Again, differentiating with respect to x, we get:

Adding equations (1) and (3), we get:

This is the required​​ differential equation of the given curve.

 

Question 6:

Form the differential equation of the family of circles touching the y-axis at the origin.

ANSWER:

The centre of the circle touching the y-axis at origin lies on the x-axis.

Let (a, 0) be the​​ centre of the circle.

Since it touches the y-axis at origin, its radius is a.

Now, the equation of the circle with centre (a, 0) and radius (a) is