The differential equation of the family of curves ${x^2} + {y^2} - 2ay = 0$ where $a$ is arbitrary constant, is

Given equation of curve is
${x^2} + {y^2} - 2ay = 0$

$\Rightarrow$ $\frac{{{x^2} + {y^2}}}{y} = 2a$

On differentiating both sides w.r.t. $x$,

we get

$\frac{{y\left( {2x + 2y\frac{{dy}}{{dx}}} \right) - \left( {{x^2} + {y^2}} \right)\frac{{dy}}{{dx}}}}{{{y^2}}} = 0$

$\Rightarrow$ $2xy + 2{y^2}\frac{{dy}}{{dx}} - \left( {{x^2} + {y^2}} \right)\frac{{dy}}{{dx}} = 0$

$\Rightarrow$ $\left( {2{y^2} - {x^2} - {y^2}} \right)\frac{{dy}}{{dx}} = - 2xy$

$\Rightarrow$ $\left( {{y^2} - {x^2}} \right)\frac{{dy}}{{dx}} = - 2xy$

$\Rightarrow$ $\left( {{x^2} - {y^2}} \right)\frac{{dy}}{{dx}} = 2xy$