NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Exercise 9.1

Q1: Determine order and degree(if defined) of differential equation 
Ans: 

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.

Q2: Determine order and degree(if defined) of differential equation 
Ans: 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.

Q3: Determine order and degree(if defined) of differential equation 
Ans: 

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.

Q4: Determine order and degree(if defined) of differential equation 
Ans: 

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.

Q5: Determine order and degree(if defined) of differential equation 
Ans: 


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.

Q6: Determine order and degree(if defined) of differential equation 
Ans: 
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.

Q7: Determine order and degree(if defined) of differential equation 
Ans: 
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.

Q8: Determine order and degree(if defined) of differential equation
Ans: 

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.

Q9: Determine order and degree(if defined) of differential equation 
Ans: 
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.

Q10: Determine order and degree(if defined) of differential equation 
Ans: 
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.

Q11: The degree of the differential equation is
(A) 3  
(B) 2  
(C) 1        
(D) not defined
Ans: 

The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.
Hence, the correct answer is D.

Q12: The order of the differential equationis
(A) 2                          
(B) 1                                      
(C) 0                          
(D) not defined
Ans: 

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

Q1: 
Ans: 

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.

Q2: 
Ans: 
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.

Q3: 
Ans: 
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.

Q4: 
Ans: 

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.

Q5: 
Ans: 
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.

Q6: 
Ans: 
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.

Q7: 
Ans: 
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.

Q8: 
Ans: 
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.

Q9: 
Ans: 
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.

Q10: 
Ans: 
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.

Q11: 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
Ans: 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.

Q12: 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
Ans: In a particular solution of a differential equation, there are no arbitrary constants.
Hence, the correct answer is D.