Which one of the following equations is a homogeneous differential eq…

The correct answer is Option (2) → \(x^2y\,dx - (x^3 + y^3)\,dy = 0\)

A differential equation \( \displaystyle\frac{dy}{dx} = F(x, y) \) is homogeneous when F is a homogeneous function of degree zero, that is, \(F(\lambda x, \lambda y) = F(x, y)\). This holds when F is a ratio of two expressions in which every term has the same degree.

Option (1) \((4x+5)\,dy + (3y-4)\,dx = 0\) — Incorrect. \( \frac{dy}{dx} = -\frac{3y - 4}{4x + 5} \); the constants 4 and 5 have degree 0, while x and y have degree 1.

Option (2) \(x^2y\,dx - (x^3 + y^3)\,dy = 0\) — Correct. \( \frac{dy}{dx} = \frac{x^2y}{x^3 + y^3} \); every term has degree 3, so \(F(\lambda x, \lambda y) = \frac{\lambda^3 x^2y}{\lambda^3(x^3 + y^3)} = F(x, y)\).

Option (3) \((x^3 + 2y^2x)\,dy + 2xy\,dx = 0\) — Incorrect. \( \frac{dy}{dx} = -\frac{2xy}{x^3 + 2xy^2} \); the numerator has degree 2 but the denominator has degree 3.

Option (4) \(x^2\,dy - y^2\,dx = \sqrt{x^2 + y^2}\,dx\) — Incorrect. \( \frac{dy}{dx} = \frac{y^2 + \sqrt{x^2 + y^2}}{x^2} \); \(y^2\) has degree 2 but \(\sqrt{x^2 + y^2}\) has degree 1.