Form the differential equation of the family of circles having centre on $y$-axis and radius 3 units.

.: Equation of any circle having centre at $y$-axis i.e., $(0,\;k)$

and radius 3 units is ${(x - 0)^2} + {(y - k)^2} = {3^2}$

$\Rightarrow {x^2} + {(y - k)^2} = 9$ …(i)

Differentiating (1) w.r.t. $x$,

we get
$2x + 2(y - k){y_1} = 0$
$\Rightarrow$ $y - k = - \cfrac{x}{{{y_1}}}$

…(2)
Substituting this value of $(y - k)$ in (1),

we get
${x^2} + {\left( { - \cfrac{x}{{{y_1}}}} \right)^2} = 9 \Rightarrow {x^2}(y_1^2 + 1) = 9y_1^2$

which is the required differential equation.