Differential Equation for Simple Inhibited Growth

The simple inhibited growth equation can be written as $$\frac{dy}{dt} = k(1 - \frac{y}{L})$$ for some constants $k$ and $L$. It can also be written as $y' = r(L-y)$, where $r = k/L$. The general solution is $$y(t) = L - c e^{-kt/L} = L - ce^{-rt},$$ where $c$ is some constant depending on the initial conditions or other data. In particular, $\lim_{t\to\infty} y(t) = L$ if $L,k,r$ are positive. In applications, $L$ is usually some sort of equilibrium value. A common application is to Newton's law of cooling.