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

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

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