$($no calculator$)$

Consider the differential equation $\frac{dy}{dx}=x^2(1-y).$

$($a$)$ On the axes provided, sketch a slope field for the given differential equation at the $9$ points indicated.

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$($b$)$ While the slope field in part $($a$)$ is drawn only at $9$ points, it is defined at every point on the $xy-$plane. Describe all points in the $xy-$plane for which the slopes are positive.

$($c$)$ Find the particular solution in the form of $y=f(x)$ to the given differential equation with the initial condition $f(0)=2$