Q.Form the differential equation representing the family of curves $x^2 + y^2 = c$, where $c$ is an arbitrary positive constant (concentric circles centred at the origin).

One differentiation eliminates the single constant directly here, with no separate substitution step needed.

Differentiating $x^2+y^2=c$ gives $2x+2y\dfrac{dy}{dx}=0$, i.e. $x+y\dfrac{dy}{dx}=0$.

The family $x^2+y^2=c$ has one arbitrary constant, $c$, so one differentiation should suffice.

Differentiating both sides with respect to $x$: $2x + 2y\dfrac{dy}{dx} = 0$ (the constant $c$, having zero derivative, disappears entirely on the right side).

Dividing through by 2: $x + y\dfrac{dy}{dx} = 0$, or equivalently $y\dfrac{dy}{dx} = -x$ — already free of $c$, with no separate elimination step required.

Check (verification, §5): take the original relation and differentiate implicitly again independently: $\dfrac{d}{dx}(x^2)+\dfrac{d}{dx}(y^2) = \dfrac{d}{dx}(c)$ gives $2x+2y\dfrac{dy}{dx}=0$, matching exactly — confirms the formed equation.

Students sometimes expect every formation problem to need a separate “solve for the constant, then substitute” step, and needlessly try to express $c=x^2+y^2$ and substitute it somewhere — but here the constant has ALREADY vanished on differentiating (since $\dfrac{d}{dx}(c)=0$ for any constant $c$), so no substitution step is needed at all; always check whether the constant has already disappeared before assuming a substitution step remains.