I have a differential equation that looks like:
$$y'=y(4-y)$$ I have identified the critical points to be $0$ and $4$. I need to identify the behavior of the equation as $t \rightarrow \infty$ and this behavior it seems depends on the initial value.
Now depending on the differential equation I can say that
1.$y > 4$, y is decreasing
2.$0<y<4$, $y$ is increasing
3.$y<0$, $y$ is decreasing
And I infer this based on the slope values calculated from the $y'$ equation.
So based on this I was able to conclude that as $t \rightarrow \infty$, $y$ converges to $4$ and and diverges from $0$. But how do I identify this behavior from intial values? I feel like I'm missing something here.