Find the differential equation of the family of all the circles touching the X axis at the origin.
Let the radius of the circle $= a$
Equation of circle touching the X axis at the origin $\left(0,0\right)$ is
$x^2 + \left(y - a\right)^2 = a^2$
i.e. $x^2 + y^2 - 2ay + a^2 = a^2$
i.e. $x^2 + y^2 - 2 a y = 0$ $\;\;\; \cdots \; (1)$
Differentiating equation $(1)$ w.r.t x gives
$2 x + 2 y \dfrac{dy}{dx} = 2 a \dfrac{dy}{dx}$
i.e. $x + y \dfrac{dy}{dx} = a \dfrac{dy}{dx}$
i.e. $a = \dfrac{x}{dy / dx} + y$ $\;\;\; \cdots \; (2)$
Substituting the value of a from equation $(2)$ in equation $(1)$ gives
$x^2 + y^2 - 2 y \left(\dfrac{x}{dy / dx} + y\right) = 0$
i.e. $x^2 + y^2 - \dfrac{2 x y}{dy / dx} - 2 y^2 = 0$
i.e. $x^2 \; \dfrac{dy}{dx} - y^2 \; \dfrac{dy}{dx} - 2 x y = 0$
i.e. $\left(x^2 - y^2\right) \dfrac{dy}{dx} = 2 x y$
is the required differential equation.