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NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Exercise 9.1

Q1: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. 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 NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is one.
It is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. The highest power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis 1. Hence, its degree is one.

Q3: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the given differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is two.
It is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential EquationsandNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. The power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis 1.
Hence, its degree is one.

Q4: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the given differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. 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 NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is two.
It is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsand the power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis 1.
Hence, its degree is one.

Q6: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is three.
The given differential equation is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations.
The highest power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis 2. Hence, its degree is 2.

Q7: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is three.
It is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. The highest power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis 1. Hence, its degree is 1.

Q8: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is one.
The given differential equation is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsand the highest power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis one. Hence, its degree is one.

Q9: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is two.
The given differential equation is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential EquationsandNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsand the highest power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis one.
Hence, its degree is one.

Q10: Determine order and degree(if defined) of differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is two.
This is a polynomial equation inNCERT Solutions Class 12 Maths Chapter 9 - Differential EquationsandNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsand the highest power raised toNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis one. Hence, its degree is one.

Q11: The degree of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis
(A) 3  
(B) 2  
(C) 1        
(D) not defined
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
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 equationNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis
(A) 2                          
(B) 1                                      
(C) 0                          
(D) not defined
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the given differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Therefore, its order is two.
Hence, the correct answer is A.

Exercise 9.2

Q1: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, differentiating equation (1) with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get the L.H.S. as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Thus, the given function is the solution of the corresponding differential equation.

Q2: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get:
L.H.S. =NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations= R.H.S.
Hence, the given function is the solution of the corresponding differential equation.

Q3: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get:
L.H.S. =NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations= R.H.S.
Hence, the given function is the solution of the corresponding differential equation.

Q4: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of the equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential EquationsL.H.S. = R.H.S.
Hence, the given function is the solution of the corresponding differential equation.

Q5: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is the solution of the corresponding differential equation.

Q6: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is the solution of the corresponding differential equation.

Q7: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations L.H.S. = R.H.S.
Hence, the given function is the solution of the corresponding differential equation.

Q8: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of the equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is the solution of the corresponding differential equation.

Q9: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is the solution of the corresponding differential equation.

Q10: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides of this equation with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value ofNCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
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.

The document NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

1. What are differential equations?
Ans. Differential equations are equations that involve an unknown function and its derivatives. They represent relationships between the function and its derivatives and are used to model various physical phenomena in science and engineering.
2. What is the order of a differential equation?
Ans. The order of a differential equation is the highest order derivative that appears in the equation. For example, the differential equation \( \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0 \) is a second-order differential equation.
3. How are differential equations classified?
Ans. Differential equations are classified based on various factors such as linearity, order, and the number of independent variables. Common classifications include ordinary differential equations (ODEs) and partial differential equations (PDEs).
4. What is the general solution of a differential equation?
Ans. The general solution of a differential equation is a solution that contains all possible solutions of the equation. It typically includes one or more arbitrary constants that can take on different values to represent different specific solutions.
5. How are initial conditions used in solving differential equations?
Ans. Initial conditions are used to find the particular solution of a differential equation that satisfies certain conditions at a specific point or points. These conditions typically involve specifying the values of the unknown function and its derivatives at a given point.
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