Form the differential equation representing the family of curves given by

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

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

$\Rightarrow 2 y^{2}=2 a x-x^{2}$                $\ldots(1)$

Differentiating with respect to x, we get:

$2 y \frac{d y}{d x}=\frac{2 a-2 x}{2}$

$\Rightarrow \frac{d y}{d x}=\frac{a-x}{2 y}$

$\Rightarrow \frac{d y}{d x}=\frac{2 a x-2 x^{2}}{4 x y}$       $\ldots(2)$

From equation (1), we get:

$2 a x=2 y^{2}+x^{2}$

On substituting this value in equation (3), we get:

$\frac{d y}{d x}=\frac{2 y^{2}+x^{2}-2 x^{2}}{4 x y}$

$\Rightarrow \frac{d y}{d x}=\frac{2 y^{2}-x^{2}}{4 x y}$

Hence, the differential equation of the family of curves is given as $\frac{d y}{d x}=\frac{2 y^{2}-x^{2}}{4 x y}$.