Logistic equation

Of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y' = y(1 - y)\\ y(0)=y_0 \end{cases}}

The solution is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = \frac{y_0e^t}{y_0 e^t - (y_0 - 1)}}

Analysis

y is undefined at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.