We have explored the following growth models, of which many can be solved by separation of variables (in the figures $y_0 = 0.1$; $r=1$; $a=4$;$b=1$ was used):

Name	Differential equation	Solution	Graph
Linear growth	$$y' = r$$	$$y(t) = r \cdot t + y_0$$
Quadratic growth	$$y'' = 2 \cdot r$$	$$y(t) = r \cdot t^2 + b \cdot t+y_0$$
Exponential growth	$$y' = r \cdot y$$	$$y(t) = y_0 \cdot e^{r \cdot t}$$
Restricted exponential growth	$$y' = r \cdot (a-y)$$	$$y(t) = a-(a-y_0)\cdot e^{-r \cdot t}$$
Gompertz growth*	$$y' = r \cdot y \cdot e^{-a \cdot t}$$	$$y(t) = y_0 e^{\frac{r}{a}\cdot (1-e^{-a \cdot t})}$$
Logistic growth	$$y'= r \cdot y \cdot (1-\frac{y}{a})$$	$$y(t) = \frac{a}{1 + (\frac{a}{y_0}-1)\cdot e^{-r \cdot t}}$$

With these models, you now have quite a set of tools to model the growth of populations, the dynamics of chemical reactions, or neural activity.

* Note that the Gompertz model is defined in different ways in the literature. Another form of the Gompertz model (for instance at Wikipedia) you might encounter is

Gompertz growth II

$$y' = r \cdot y \cdot \ln(\frac{a}{y})$$

$$y(t) = a \left(\frac{a}{y_0} \right)^{-e^{-r \cdot t}}$$ $$= a^{1-e^{-r \cdot t}}\cdot y_0^{e^{-r \cdot t}}$$