If the differential equation representing the family of all circles touching x - axis at the origin is $\;(x^{2}-y^{2})\large\frac{dy}{dx}=g(x) y\;$, then $\;g(x)\;$ equals :

Question

$(a)\;\large\frac{1}{2} x\qquad(b)\;2x^{2}\qquad(c)\;2x\qquad(d)\;\large\frac{1}{2} x^{2}$