Editing Separation of variables (section)

=== Example ===
Population growth is often modeled by the "logistic" differential equation

: <math>\frac{dP}{dt}=kP\left(1-\frac{P}{K}\right)</math>

where <math>P</math> is the population with respect to time <math>t</math>, <math>k</math> is the rate of growth, and <math>K</math> is the [[carrying capacity]] of the environment.
Separation of variables now leads to

: <math>
\begin{align}
& \int\frac{dP}{P\left(1-P/K \right)}=\int k\,dt
\end{align}
</math>
which is readily integrated using partial fractions on the left side yielding

: <math>P(t)=\frac{K}{1+Ae^{-kt}}</math>

where A is the constant of integration. We can find  <math>A</math> in terms of <math>P\left(0\right)=P_0</math> at t=0. Noting <math>e^0=1</math> we get

: <math>A=\frac{K-P_0}{P_0}.</math>