Given $y'=3y$ an dynamical system. I'm asked to find the time elapsed between $y=5$ and $y=185$. As $$\int_{T_{a}}^{T{b}} dt$$ definition suggests, I've calculated $$\int_{5}^{185} dt = \int_{5}^{185} \frac{dy}{3y} = \frac{1}{3} (ln|185|-ln|5|)$$ However, the question continues with the part that asks calculation of the time elapsed between $y=-50$ and $y=600$. Similarly, as $$\int_{-50}^{600} dt = \int_{-50}^{600} \frac{dy}{3y} = \frac{1}{3} (ln|600|-ln|-50|) = \frac{1}{3} (ln600-ln50)$$
Firstly, I wonder whether I made any mistake. I hardly grasp the notion of time elapsed between phase path points. The critical point here is $y=0$. Thus, the phase portrait is simply y-axis with $y=0$ as an unstable fixed point. What is the difference between these two calculations? Can you explain? Thanks in advance.
In the second case you are right to be doubtful, as you can wait for all times, but starting at negative value will never lead to positive values.