The logistic differential equation $y'=y(b-ay) \, \textrm{with}\, a\neq 0, b\neq 0$ has the non-trivial solution $y(t) = \frac{\frac{b}{a}}{1+e^{-bt}}, \quad (1)$ where $c$ is a constant.
My questions is:
We have that $\frac{y'}{y}= b-ay$. But how can I understand this intuitively. What does it give?