Q1:
Ans: The given differential equation i.e., (x^{2} xy) dy = (x^{2} y^{2}) 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) = λ^{n} 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.