NCERT Solutions - Exercise 9.4: Differential Equations

# NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q1:
Ans: The given differential equation i.e., (x2   xy) dy = (x2   y2) dx can be written as:

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:
Substituting the values of v and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q2:
Ans: The given differential equation is:

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:
Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q3:
Ans: The given differential equation is:

Thus, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q4:
Ans: The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q5:
Ans: The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution for the given differential equation.

Q6:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the values of and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q7:
Ans: The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q8:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q9:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

Therefore, equation (1) becomes:

This is the required solution of the given differential equation.

Q10:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vy

Substituting the values of x and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Q11:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

Now, y = 1 at x = 1.

Substituting the value of 2k in equation (2), we get:

This is the required solution of the given differential equation.

Q12:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: = vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

Now, y = 1 at x = 1.

Substituting in equation (2), we get:

This is the required solution of the given differential equation.

Q13:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve this differential equation, we make the substitution as:
= vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

Now, .

Substituting C = e in equation (2), we get:

This is the required solution of the given differential equation.

Q14:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the values of y and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.
Now, y = 0 at x = 1.

Substituting C = e in equation (2), we get:

This is the required solution of the given differential equation.

Q15:
Ans:

Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
= vx

Substituting the value of y and in equation (1), we get:

Integrating both sides, we get:

Now, y = 2 at x = 1.

Substituting C = –1 in equation (2), we get:

This is the required solution of the given differential equation.

Q16: A homogeneous differential equation of the form can 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 form, 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.
B.
C.
D.
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:

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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## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## 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.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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