Find the differential equation for the family of curves $y^{2}=a(b^{2}-x^{2})$ by eliminating the arbitrary constants $a$ and $b$.

(D) Given equation: $y^{2}=a(b^{2}-x^{2})$
Differentiating both sides with respect to $x$:
$2y \frac{dy}{dx} = a(-2x)$
$y y' = -ax$ --- $(1)$
Differentiating again with respect to $x$:
$y' y' + y y'' = -a$
$(y')^{2} + y y'' = -a$ --- $(2)$
From $(1)$,we have $a = -\frac{y y'}{x}$. Substituting this into $(2)$:
$(y')^{2} + y y'' = -(-\frac{y y'}{x})$
$(y')^{2} + y y'' = \frac{y y'}{x}$
Multiplying by $x$:
$x(y')^{2} + x y y'' = y y'$
$x y y'' + x(y')^{2} - y y' = 0$