JEE Exam  >  JEE Notes  >  Mathematics (Maths) Class 12  >  NCERT Solutions - Exercise 9.4: Differential Equations

NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q1: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation i.e., (x2   xy) dy = (x2   y2) dx can be written as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This shows that equation (1) is a homogeneous equation.
To solve it, we make the substitution as: = vx
Differentiating both sides with respect to x, we get: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of v and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q2: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Thus, the given equation is a homogeneous equation.
To solve it, we make the substitution as: = vx
Differentiating both sides with respect to x, we get: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q3: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Thus, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q4: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q5: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution for the given differential equation.

Q6: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q7: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q8: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q9: 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
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, equation (1) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q10: 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
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vy
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of x and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q11: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, y = 1 at x = 1.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value of 2k in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q12: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, y = 1 at x = 1.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q13: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve this differential equation, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting C = e in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q14: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.
Now, y = 0 at x = 1.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting C = e in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q15: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, y = 2 at x = 1.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting C = –1 in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required solution of the given differential equation.

Q16: A homogeneous differential equation of the form NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationscan be solved by making the substitution
A. y = vx 
B. v = yx
C. = vy 
D. x = v
Ans: For solving the homogeneous equation of the formNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations, we need to make the substitution as x = vy.Hence, the correct answer is C.

Q17: Which of the following is a homogeneous differential equation?
A. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations                   
B. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
C. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations                                   
D. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: Function F(x, y) is said to be the homogenous function of degree n, if
F(λx, λy) = λn F(x, y) for any non-zero constant (λ).
Consider the equation given in alternativeD:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the differential equation given in alternative D is a homogenous equation.

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

1. What is a differential equation?
Ans. A differential equation is an equation that relates one or more functions and their derivatives. It describes the relationship between a function and its rate of change.
2. How are differential equations classified?
Ans. Differential equations can be classified as ordinary differential equations (ODEs) or partial differential equations (PDEs) based on the number of independent variables they involve.
3. What is the order of a differential equation?
Ans. The order of a differential equation is the highest order derivative present in the equation. For example, if the equation involves the second derivative of a function, it is a second-order differential equation.
4. How are initial conditions and boundary conditions used in solving differential equations?
Ans. Initial conditions are used for solving initial value problems, where the solution needs to satisfy certain conditions at a specific point. Boundary conditions are used for solving boundary value problems, where the solution needs to satisfy certain conditions over a specific interval.
5. What are some common methods used to solve differential equations?
Ans. Some common methods used to solve differential equations include separation of variables, integrating factors, substitution methods, and using power series solutions. Different methods are applied based on the type and complexity of the differential equation.
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