The differential equation of the family of circles passing through the fixed points $(a, 0)$ and$(- a, 0),$ is 

Let the equation of the family of circles be

$x^2+y^2+2gx+2fy+c=0$

It passes through $(a, 0)$and $(- a, 0)$. Therefore, 

$a^2+2ag+c=0$ and $a^2-2ag+c=0$

Solving these two equations, we get $c=-a^2$ and $g=0$

Substituting the values of c and gin (i), we get

$x^2+y^2+2fy-a^2=0$

It is a one parameter family of circles. 

Differentiating with respect to x, we get 

$2x+2y{y}_1+2f{y}_1=0\Rightarrow f=-\left(\frac{x+y{y}_1}{y_1}\right)$

Substituting the value off in (ii), we get

$y_1\left(y^2-x^2+a^2\right)+2xy=0$

This is the required differential equation