A cup of coffee cooling on a desk and a sample of radioactive material decaying in a lab have almost nothing physically in common — different substances, different mechanisms, different scales entirely. Both, remarkably, are governed by the exact same mathematical form: a differential equation stating that the rate of change of some quantity is proportional to the quantity itself. That single relationship, once solved, predicts both systems' entire future.
Describing change, not the thing itself
An ordinary equation like y = x² tells you a value directly. A differential equation instead relates a quantity to its own rate of change — its derivative — describing not what the quantity is, but how it's currently behaving, and letting that rule of behaviour determine everything that follows. For both cooling and radioactive decay, that rule takes the same simple form: the rate of change is proportional to the current amount. A hot object loses heat faster when it's further above the surrounding temperature (Newton's law of cooling), and radioactive material decays at a rate proportional to how much undecayed material remains (the law of radioactive decay). Different physical mechanisms, identical mathematical structure.
Solving the equation reveals the exponential curve underneath both
Solving this type of differential equation produces an exponential function — for cooling, the temperature difference between the object and its surroundings shrinks exponentially over time, meaning it drops quickly at first and then more and more slowly, mathematically never quite reaching zero even though it becomes practically indistinguishable from it. Radioactive decay follows the identical exponential curve, which is why radioactive decay is conventionally described using "half-life" — the fixed time it takes for exactly half of any remaining amount to decay, regardless of how much was there to begin with, a property that follows directly and necessarily from the underlying exponential mathematics rather than being a separate, additional rule. The same equation, in different contexts, also describes population growth under unlimited resources, the discharge of a capacitor in an electrical circuit, and the buildup of a drug's concentration in the bloodstream — a strikingly wide range of real systems, all reducible to the same underlying differential equation.
What we're still unsure about
This particular differential equation — rate of change proportional to current amount — is one of the simplest kinds and has a clean, exact, well-understood solution; it isn't a subject of dispute. Most differential equations that arise from real, complex systems, however, don't have a clean closed-form solution at all, and have to be solved approximately using numerical methods on a computer instead. Predicting the long-term behaviour of far more complicated differential equations — including some describing weather systems, fluid turbulence, and other chaotic processes — remains a genuinely hard, actively researched problem in applied mathematics, where tiny differences in a starting condition can produce wildly different outcomes that no amount of computational power fully tames.
This sits inside Differential Equations, one of eight topics in Calculus, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.