The one sentence every model is hiding
By now you can solve a differential equation once someone hands it to you — separate it, find an integrating factor, check the answer by substitution. But out in the world nobody hands you an equation. They hand you a paragraph: 'a colony of bacteria doubles every three hours', 'a cup of coffee cools toward room temperature', 'brine flows into a tank while the mixture drains out'. The real skill this rung is built on is the translation itself — turning English into a differential equation — and it is a craft you can learn deliberately rather than guess at.
The secret is that almost every such paragraph is paraphrasing a single template: 'the rate of change of some quantity equals some expression'. Hunt for the word 'rate', or 'per', or 'grows', or 'cools', or 'flows' — each is a disguised derivative. If Q is the quantity, the sentence is really saying dQ/dt = (something). The whole job is to identify Q, recognise that the changing-language means dQ/dt, and then express the 'something' in terms of Q, t, and the constants of the problem.
Naming the cast: variables and what they mean
Before any symbols appear, decide explicitly what each one stands for, including its units. This is not bureaucracy; it is where most modeling errors are quietly born. Choose the independent variable — usually time t in seconds, hours, or years — and the dependent variable, the unknown function you are tracking: a temperature T(t) in degrees, a mass m(t) in grams, a count N(t) of animals. Writing 'let T(t) be the coffee's temperature in degrees Celsius at time t minutes after pouring' forces you to be honest about which variable depends on which.
Units double as a free error-checker. The left side dQ/dt always carries units of (quantity per time): grams per minute, degrees per second. So every term on the right must carry those same units too. If your right side comes out as 'grams' with no 'per minute', you have dropped a rate constant or mis-set the equation — the mismatch catches the mistake before you ever try to solve. This habit of balancing units is the cheapest insurance in all of modeling.
From law to equation: three worked translations
Watch the template at work. Growth. 'A population grows at a rate proportional to its current size.' Quantity P, rate dP/dt, the phrase 'proportional to its size' means equal to a constant times P. So dP/dt = k P — the law of exponential growth. The constant k is the per-capita growth rate; its sign and size you will later read off real data.
Cooling. 'A hot object cools at a rate proportional to how much hotter it is than the room.' Quantity T, rate dT/dt, and 'how much hotter than the room' is the difference T - T_room. So dT/dt = -k (T - T_room), with a minus sign because hot things cool *down* — this is Newton's law of cooling. Mixing. 'Salt water flows into a tank and the mixture drains out.' Quantity = grams of salt S, and dS/dt = (rate salt enters) - (rate salt leaves), a balance of two flows you will assemble carefully in a later guide.
English phrase Differential equation ---------------------------------- --------------------------- rate proportional to amount dP/dt = k P rate proportional to (T minus room) dT/dt = -k (T - T_room) inflow rate minus outflow rate dS/dt = r_in - r_out rate that slows as P nears a ceiling dP/dt = k P (1 - P/K)
Notice the last row of the table. When a problem adds 'but the growth slows as the population approaches a ceiling K', you multiply the growth law by a braking factor (1 - P/K) that fades to zero at the carrying capacity. That is the logistic equation, and it shows the craft scaling up: complications in the story become extra factors and terms on the right-hand side, never a change to the basic template.
Tidying up: normal form and the starting value
Once the equation exists, put it in first-order normal form — solved for the derivative, dy/dt = f(t, y). This is more than tidiness: nearly every solving recipe and every numerical method assumes the equation is in this shape, because f(t, y) is precisely the slope the solution must have at the point (t, y). Getting there is just algebra — divide through, move terms — but it is the standard doorway every later technique walks in through.
The equation alone still hides a whole family of solutions, one for each value of the constant. The word problem almost always supplies the missing fact: 'the coffee starts at 90 degrees', 'there are initially 500 fish'. That single starting fact is an initial condition like y(0) = 90, and the equation together with it forms an initial value problem — the precise mathematical object that has, under mild assumptions, exactly one solution. A model without its initial condition is only half a model.
- Name the variables and their units; state what depends on what — 'let T(t) be temperature in degrees at time t minutes'.
- Find the verb of change. Whatever 'grows', 'cools', or 'drains' becomes the derivative dQ/dt on the left.
- Build the right side from the law: translate 'proportional to', 'in minus out', or 'slows near a ceiling' into Q, t, and constants. Check units balance.
- Rearrange into normal form dQ/dt = f(t, Q), then attach the initial condition Q(0) = Q_0 to lock onto one solution.
Honesty about what a model is
Translating a paragraph is only one step of a longer loop. The full modeling cycle runs: simplify reality into assumptions, write the equation, solve or analyse it, then compare the prediction against actual data and circle back to fix the assumptions when it disagrees. A differential equation is never the truth; it is a deliberate, simplified caricature of the truth, useful exactly to the extent that its assumptions hold.
So name your assumptions out loud and respect their limits. 'dP/dt = k P' quietly assumes unlimited food and space — true for a young culture, absurd once the dish fills, which is exactly why the logistic correction exists. 'dT/dt = -k(T - T_room)' assumes the room's temperature never budges and the object has one uniform temperature, both only approximately so. Every model has a domain of honesty; pretending it holds everywhere is how good mathematics produces bad predictions.