Differential equations · 01 · Nature's native language · 8 min
Reading rate laws
Nature almost never tells you what a quantity is — it tells you how the quantity changes. A differential equation is that sentence written in symbols, and learning to read it aloud is half the subject.
Build the intuition
The unknown is a whole curve
In algebra, x is one number. In y′ = ky, the unknown y is an entire function — the full story of a quantity over time. Solving means finding the curve whose slope, at every moment, obeys the law. New kind of question, new kind of answer.
Translate before you solve
y′ = ky reads: “the more there is, the faster it grows” (money, bacteria). T′ = −k(T − 20) reads: “coffee cools faster the hotter it is than the room.” v′ = g − cv: “gravity pulls, drag pushes back harder at speed.” Translating to plain English is genuinely most of the work.
One law, many curves — until you pin the start
A rate law constrains the shape of solutions but not the starting point: y′ = ky is satisfied by every curve Ce^{kt}. Add an initial condition — “at t = 0 there were 100 bacteria” — and exactly one curve survives. Law + starting point = prediction. (You met this as +C in antiderivatives.)
See it move
The law plants a slope at every point of the plane. Read the field first; the solution simply obeys it.
A worked example
Translate three laws
A savings account with 5% continuous interest:
A cooling coffee in a 20° room:
A skydiver under gravity and air drag:
Each is a sentence about change. None tells you M, T, or v directly — solving will.
Out in the world
F = ma is a differential equation
Newton's law looks algebraic but acceleration is the second derivative of position: F = m·x″. Every orbit, trajectory, and crash-test simulation starts by writing this rate law for the forces involved — physics is differential equations wearing a famous costume.
Common confusion, cleared
“Solving means finding a number.”
The solution is a function — the whole future (and past) of the quantity. y(t) = 100e^{0.05t} answers every “when?” at once.
“The equation hides the behavior until you solve it.”
Often the law itself talks: y′ = ky with k > 0 must explode; T′ = −k(T−20) must settle at 20. Reading qualitative behavior straight from the law is a professional skill — and beginners can start immediately.
Recap
- Differential equations state how things change; the unknown is a function.
- Translate symbols to sentences first — that's reading the law.
- Law + initial condition = one unique predicted curve.