Take the 2-minute tour ×  The logistic differential equation
$$y'=y(b-ay) \, \textrm{with}\, a\neq 0, b\neq 0$$ has the non-
trivial solution $$y(t) = \frac{\frac{b}{a}}{1+c\cdot e^{-bt}},
\quad (1)$$ where $c$ is a constant.  My question is: If
$0<y<\frac{b}{a}$, then we get the s-form curve. If
$y>\frac{b}{a}$ then we get the decreasing graph which is above
it. In which situations do we get this curve. Can you give an
example. We say that $y>0$, but when do we have that $y$ is also
a solution?  share|improve this question I have seen but this
has not been asked before. –  Reader Nov 14 '12 at 4:27 add
comment  1 Answer  up vote 1 down vote accepted  You can see the
behavior of the solutions even without using the explicit
solution. The equation is $y'=y(b-a y)$. I'll assume in fact
that $a,b>0$ rather than just that they are nonzero; easy
variation if not.  Now make a "slope field" by doing the
following: At each point $(x,y)$ in the plane (or really onoly
at a lot of points) make a tiny line segment having