Drug Assimilation Into the Blood

When a pill is swallowed, its ingredients don't reach the bloodstream instantly — they first dissolve in the gastrointestinal tract (GI tract) and diffuse from there into the blood, from where the body's tissues absorb them and the kidneys and liver eventually remove them, each at its own rate. Studying how quickly a drug is assimilated, and how long it stays active, is itself an application of differential equations.

Consider the simplest case: a single dose is taken, with no further doses afterward. Let $x(t)$ be the amount of drug still present in the GI tract at time $t$, with $x(0) = x_0$ the amount taken initially. Since no more drug enters the GI tract after the dose is taken, the only thing changing $x$ is the rate at which drug leaves the GI tract to enter the blood — and that outflow rate is proportional to how much drug is still there to diffuse: