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 to
is 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 in
and
. The power raised to
is 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 in
and the power raised to
is 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 to
is 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 to
is 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 in
and the highest power raised to
is 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 in
and
and the highest power raised to
is 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 in
and
and the highest power raised to
is 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 equation
is
(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 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.
Q2: 
Ans: 
Differentiating both sides of this equation with respect to x, we get:
Substituting the value of
in 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 of
in 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 of
in 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 of
in 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 of
in 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 of
in 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 of
in 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.
