The differential equation \(\displaystyle\frac{dy}{dx} = \frac{x^n + y^3}{xy^2}\) is a homogeneous differential equation when n is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The equation \(\frac{dy}{dx} = F(x, y)\) is homogeneous when F is a homogeneous function of degree zero.
The denominator \(xy^2\) has degree 3, and the term \(y^3\) has degree 3. For the whole fraction to have degree zero, \(x^n\) must also have degree 3:
\( n = 3 \)
Check with n = 3:
\( \displaystyle F(\lambda x, \lambda y) = \frac{\lambda^3 (x^3 + y^3)}{\lambda^3 xy^2} = F(x, y) \)
For any other n, the numerator is not homogeneous, because its two terms have different degrees.