Expii

# 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.