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Quiz 5 solution
Find and classify the equilibrium points of $y'=y^2+y-2$.
Solution: First find the critical points by solving $$0=y^2+y-2=(y+2)(y-1),$$ which has solutions $y=1,-2$. This partitions the $t$-axis into three intervals: $(-\infty,-2)$, $(-2,1)$, $(1,\infty)$. We pick a "test point" in each interval, say $y_1=-3$, $y_2=0$, $y_3=2$. Now we evaluate the right-hand side of the differential equation at each of these test points:

test point $y_i$	$y_i^2+y_i-2=$
$-3$	$4 > 0$
$0$	$-2 < 0$
$2$	$4 > 0$

From this data, we are able to conclude that the equilibrium solution $y=-2$ is asymptotically stable and the equilibrium solution $y=1$ is unstable.