The One Equation Behind Three Phenomena
In the previous guide you practised turning a sentence into a derivative — going from a word problem to a differential equation. Now we cash in. The most common sentence in all of modeling is this: *the rate of change of a quantity is proportional to the quantity itself.* Written down, that is simply y' = k y, where k is a constant. One short equation, yet it governs growing bank balances, splitting bacteria, decaying atoms, and cooling fluids. Learn it once and you have learned them all.
Why is proportionality so natural? Picture a colony of bacteria. Each cell divides on its own schedule, blind to the others. Double the number of cells and you double the number dividing per minute — so the growth rate scales with the population. The same logic runs backward for decay: each radioactive atom has a fixed chance of breaking apart this second, independent of its neighbours, so the number decaying per second is proportional to how many are left. Whenever the members of a population act independently, you get y' = k y almost for free.
This is a separable and also a first-order linear equation, so you already own two ways to solve it. Separating gives (1/y) dy = k dx, and integrating delivers the famous answer y(t) = y0 e^(kt), where y0 is the starting amount. Everything hangs on the sign of k: if k > 0 the exponential climbs (growth); if k < 0 it falls toward zero (decay). That single sign is the difference between a population explosion and a fading radioactive sample.
Reading the Constant: Doubling Time and Half-Life
The constant k carries units of 1/time and is a bit abstract. Scientists prefer a number you can feel. For growth, that is the doubling time — how long until the quantity doubles. Set e^(k T) = 2 and you find T = ln(2)/k, a fixed interval no matter where you start: a culture that doubles every 20 minutes does so whether it has a hundred cells or a billion. For decay, the mirror-image number is the half-life: solve e^(k T) = 1/2 (with k negative) to get T = ln(2)/|k|, the time for half the sample to vanish.
This is the engine of carbon dating. Living things absorb carbon-14 while alive; once they die the intake stops and the carbon-14 decays with a half-life of about 5,730 years. Measure how much is left, run the formula y(t) = y0 e^(kt) backward, and you read off the age. Radioactive decay is the cleanest real-world example of y' = k y there is, because the independence assumption is not an approximation — quantum mechanics really does make each atom decay on its own, with no memory and no influence from its neighbours.
When the Target Isn't Zero: Newton's Cooling
Pure growth and decay aim at either infinity or zero. But a hot coffee on the counter does neither — it cools toward room temperature and then stops. We need a model whose target is a non-zero number. Newton's law of cooling supplies it with a small, beautiful twist: the rate of cooling is proportional not to the temperature itself, but to the *difference* between the object and its surroundings. If T is the object's temperature and M the ambient temperature, then T' = -k (T - M), with k > 0.
Look at what the difference does. When the coffee is much hotter than the room, T - M is large, so it cools fast. As T sinks toward M, the difference shrinks, and the cooling slows to a crawl — which matches your experience exactly: a scalding cup loses heat quickly at first, then lingers near lukewarm for ages. When T reaches M, the right side is zero, T' = 0, and the temperature stops. That resting value T = M is an equilibrium solution: place the object there and it never moves.
Here is the lovely shortcut. Let u = T - M, the *excess* temperature above the room. Since M is constant, u' = T', and the equation becomes u' = -k u — exactly the decay equation you already mastered! So u(t) = u0 e^(-kt), and translating back, T(t) = M + (T0 - M) e^(-kt). The gap between object and room simply decays exponentially to zero; the temperature itself decays exponentially to M. Cooling is just decay with the target slid up from 0 to M.
decay : y' = k y -> y(t) = y0 e^(kt) target 0
cooling : T' = -k (T - M) -> T(t) = M + (T0 - M) e^(-kt) target M
substitution u = T - M turns the second line into the first:
u' = -k u -> u(t) = u0 e^(-kt)Transient and Steady: The Time Constant
The cooling solution T(t) = M + (T0 - M) e^(-kt) has two parts that are worth naming, because they recur everywhere in first-order linear models. The constant M is the steady state — where the system settles. The fading piece (T0 - M) e^(-kt) is the transient — the memory of the initial condition that dies away. This split is the whole idea of transient plus steady state: every such system forgets its start exponentially and is left with only its destination.
How long does the transient last? Engineers answer with the time constant tau = 1/k. It is the time for the gap to shrink to 1/e — about 37% — of its starting size. After one tau you are 63% of the way to room temperature; after about 3 tau you are within 5%; after 5 tau the difference is under 1% and, for any practical purpose, the system has arrived. The time constant is the single number that tells you 'how fast', just as the half-life did for decay — indeed tau and the half-life are the same idea wearing different clothes, related by half-life = tau · ln(2).
Where These Models Bend and Break
Every good modeler keeps a list of the limits of the model, and these three have famous ones. Unbounded exponential growth is the least honest: y' = k y says a bacterial colony fills the ocean in a week and the universe shortly after. No real population does this, because food, space, and waste eventually push back. The fix is to let k depend on y — and that is exactly the road to the logistic equation in the next guide, where growth bends over toward a carrying capacity.
Newton's cooling is honest but only approximate. It assumes the object has a single uniform temperature and that k is constant — neither is perfectly true. A real roast has a hot core and a cooler crust, so it does not cool as one lumped number; and at large temperature differences, radiation (which scales like the fourth power of absolute temperature, not the first) takes over and the simple linear law underestimates the loss. Newton's law is excellent for modest temperature gaps and gentle convection, and it quietly degrades outside that range. Knowing *where* a model is trustworthy is as valuable as the model itself.
There is also a quieter subtlety from your separation-of-variables training. Solving y' = k y by separating divides by y, which silently drops the constant solution y = 0 — a genuine lost solution. For growth and decay this rarely bites, since y = 0 just means 'nothing was there', and the exponential family already approaches it. But the habit matters: the same casual division costs you real equilibria in the logistic model ahead. The honest exponential answer is y = y0 e^(kt) *together with* the equilibrium y = 0, even if the latter usually goes without saying.