Which one of the following equations is a homogeneous differential equation?

$x^2y\, dx-(x^3+ y^3)\,dy = 0$

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

Given Question: Which one of the following equations is a homogeneous differential equation?

Option 1: (4x + 5) dy + (3y − 4) dx = 0

→ Not homogeneous, since 4x + 5 and 3y − 4 are not homogeneous expressions of the same degree.

Option 2: x²y dx − (x³ + y³) dy = 0

→ All terms are of degree 3: x²y, x³, y³ → homogeneous of degree 3.

This is a homogeneous differential equation.

Option 3: (x³ + 2y²x) dy + 2xy dx = 0

→ x³ and 2y²x are degree 3, 2xy is degree 2 → not all terms same degree → Not homogeneous

Option 4: x² dy − y² dx = √(x² + y²) dx

→ RHS involves a square root; not polynomial → Not homogeneous