The differential equation of the family of curves \(y = cx^3\), where c is an arbitrary constant, is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(x\displaystyle\frac{dy}{dx} = 3y\)
The family has one arbitrary constant, so its differential equation is of order 1, obtained by eliminating c.
Differentiating \(y = cx^3\):
\( \displaystyle \frac{dy}{dx} = 3cx^2 \)
From the family, \(c = \frac{y}{x^3}\). Putting this in:
\( \displaystyle \frac{dy}{dx} = 3 \cdot \frac{y}{x^3} \cdot x^2 = \frac{3y}{x} \)
\( \displaystyle x\frac{dy}{dx} = 3y \)
Option (2) still describes only the curve with c = 1 (and its shifts up and down), because the constant has not been eliminated.