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 inand
and 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 of
in 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 of
in 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+2bx⇒y'-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