Logistic equation

Of the form

${\begin{cases}y'=y(1-y)\\y(0)=y_{0}\end{cases}}$

The solution is

$y(t)={\frac {y_{0}e^{t}}{y_{0}e^{t}-(y_{0}-1)}}$

Analysis

y is undefined at $t=\ln \left({\frac {y_{0}-1}{y_{0}}}\right)$

If 0 < y_0 < 1, The solution is defined for all t. y(t) approaches 1 as t approaches infinity. y(t) approaches 0 as t approaches negative infinity.

For y_0 > 1, the blowup time is negative

$\ln \left({\frac {y_{0}-1}{y_{0}}}\right)<0$

As t approaches positive infinity, y(t) approaches 1.

For y_0 < 0, as t approaches negative infinity, y(t) approaches 0.