Differential Equation for Simple Inhibited Growth - Expii

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.