Q.Form the differential equation representing the family of curves $y = cx^2$, where $c$ is an arbitrary constant.

One arbitrary constant means one differentiation is enough; eliminate $c$ using the original relation.

Differentiating gives $\dfrac{dy}{dx}=2cx$; since $c=y/x^2$, substituting gives $x\dfrac{dy}{dx}=2y$.

The family $y=cx^2$ has exactly one arbitrary constant, $c$, so one differentiation should be enough to eliminate it.

Differentiating with respect to $x$: $\dfrac{dy}{dx} = 2cx$.

From the original relation, $c = \dfrac{y}{x^2}$ (valid for $x\neq0$).

Substituting this into the derivative equation: $\dfrac{dy}{dx} = 2x\cdot\dfrac{y}{x^2} = \dfrac{2y}{x}$.

Rearranging to clear the fraction: $x\dfrac{dy}{dx} = 2y$, which is free of $c$ — the required first-order differential equation.

Check (verification, §5): take the general solution $y=cx^2$, differentiate to get $\dfrac{dy}{dx}=2cx$, and substitute both back into the formed equation: $x(2cx) = 2cx^2 = 2(cx^2) = 2y$ — the identity holds for every $c$, confirming the formed differential equation is correct.

A common slip is stopping after differentiating once, leaving the equation as $\dfrac{dy}{dx}=2cx$ — this is not yet the answer, since $c$ is still present. The constant must always be eliminated by substitution using the original relation; a formed differential equation must never contain the arbitrary constant it was formed to eliminate.