The general solution of the differential equation \(\displaystyle\frac{dy}{dx} = -\frac{2x}{3y^2}\), \(y \ne 0\), is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(x^2 + y^3 = C\)
The equation is variables separable:
\(3y^2\,dy = -2x\,dx\)
Integrating both sides:
\(y^3 = -x^2 + C\)
\(x^2 + y^3 = C\)
Check by differentiating: \(2x + 3y^2\,\frac{dy}{dx} = 0\), which gives \(\frac{dy}{dx} = -\frac{2x}{3y^2}\).