NCERT Solutions - Exercise 9.4: 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: y = 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: y = 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:
y = 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: y = 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: y = 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:
y = vx

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.

Q7: 
Ans: The given differential equation is:


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