${y^2} = a({b^2} - {x^2})$

.: We have, ${y^2} = a({b^2} - {x^2})$

…(1)
Differentiating (1) w.r.t. $x$,

we get
$2y{y_1} = a( - 2x) \Rightarrow y{y_1} = - ax$
…(2)
Again differentiating (2) w.r.t. $x$,

we get
$y{y_2} + {y_1}{y_1} = - a$ …(3)
Eliminating $a$ between (2) and (3),

we get
$y{y_1} = (y{y_2} + {y_1}{y_1})x \Rightarrow y{y_1} = x(y_1^2 + y{y_2})$

which is the required differential equation.