Q.Find the differential equation of the curve $x^2 + y^2 = a^2$

$x^2+y^2=a^2$ contains one arbitrary constant, so one differentiation eliminates it: $x+y\dfrac{dy}{dx}=0$.

The family of curves $x^2 + y^2 = a^2$ (concentric circles centred at the origin) has a single arbitrary constant $a$. To form the differential equation, differentiate with respect to $x$:

$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2)$

$2x + 2y\frac{dy}{dx} = 0.$

Dividing throughout by $2$:

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