Separable equations

Level UniversityDifficulty ★★★★★Method⌖ Open in the map

What is it?

$y' = g(t)\,h(y)$ : put all the $y$ on one side and integrate, $\int\frac{\dd y}{h(y)} = \int g(t)\,\dd t$ . Exponential growth and the logistic curve are solved this way.

Formulas

$y' = r\,y\Big(1 - \frac{y}{K}\Big) \implies y(t) = \frac{K}{1 + \big(\frac{K}{y_0} - 1\big)e^{-rt}}$
logistic growth: the sigmoid is its solution

Where it shows up in computing

Population and epidemic models★★★★★frequentPhysics and simulation
The logistic equation of population and adoption curves is separable; its solution is a shifted sigmoid.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

Population and epidemic models★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.