$($a$)$ Show that $y^{2}x-6=0$ is an implicit solution to $\frac{dy}{dx}=-\frac{1}{2y}$ on the interval $(-,6)$

$($b$)$ Show that $x{y}^3-x{y}^{3}cosx=1$ is an implicit solution to the differential equation $\frac{dy}{dx}=\frac{(-xsinxcosx-1)y}{3(x-xcosx)}$ on the interval $(0,2).$

$($a$)$ To verify the solution and the interval, convert $y^{2}x-6=0$ to explicit function$($s$).$

$y=($Use a comma to separate answers as needed.$)$