Which one of the following equations is a homogeneous differential equation?
Answer & explanation
Correct answer: option 2
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