Given the Cauchy Problem,
$$ y'(x) = \arctan[(2-y^2)(x^2+xy)] $$ with initial conditions $y(0)=k$
I know that there are two horizontal asymptotes at $y=+\sqrt2$ and $y=-\sqrt2 $. Also, the $y'=0$ at $y=-x$. But I cannot understand how the slope increases or decreases between these asymptotes. The below-attached graph indicates when the slope increases (denoted as $+$) and the slope decreases (denoted as $-$). May I know how these signs are evaluated?
