Consider the differential equation

$\frac{dy}{dx}+2x{y}^2=0$

$($a$)(2$ points$)$ Is this equation linear or nonlinear?

$($b$)(6$ points$)$ Show that $y=\frac{1}{x^2-1}$ is a solution to the given differential equation satisfying $y(0)=-1.$

$($c$)(2$ points$)$ What is the interval of existence for this solution $($i$.$e$.$ on what interval is this solution valid$)?$