You don't need calculus to understand this. You just need one idea: sometimes we know how fast
something is changing right now, without knowing the whole story of what it'll do next.
A differential equation is simply that idea, written down as a rule. This page lets you use that
rule to build the future yourself, one small step at a time.
Think of a speedometer. It only ever tells you your speed right now — your car might be slowing down or speeding up at different rates. Sometimes you put the pedal to the metal and sometimes you slam on the brakes. You might even notice the gas pedal seems to work differently when you change gears. However, if you are interested in distance, you could use many speedometer measurements at various points in time to show how your speed affects the distance you travel, even when you are driving erratically creatively.
The rule
Time elapsed0.0 s
Value—
Rate right now—
Steps taken0
Building the curve, one step at a time
What just happened
Every time you clicked Step, the page did exactly three things: it looked at the rule to
find the current rate of change, it moved forward a small distance (Δt) in that direction,
and it landed on a new point — which became the new "right now." Repeat that enough times and a
whole curve appears, even though at no point did anyone tell the page the full shape in advance.
That's all a differential equation is: a rule for the rate of change, written in terms of the
current state, instead of a formula for the state itself. "Solving" one means walking that rule
forward through time. Make Δt smaller and each step gets more honest — the jagged path
tightens into a smooth curve. This is precisely what was happening, invisibly, 60 times a second,
inside the PET reactor simulator: it's the same stepping idea, just with a smarter, more accurate step
(called Runge–Kutta) so it needs far fewer of them to stay accurate.
So, what is a differential equation used for?
A differential equation is a rule that tells you how fast something changes right now,
based on how much of it there currently is — not a formula for the thing itself.
To find out what happens next, you follow that rule forward, a tiny step at a time.
You now know what a differential equation is: a rate rule, not a value formula.
You now know what "solving" one means: stepping the rule forward through time, from a known starting point.
You now know why step size matters: smaller steps trade speed for accuracy — the exact tradeoff every real ODE solver has to manage.
You now know why some of these can't be solved with a formula at all (like PET breakdown) — and why that's exactly when stepping methods matter most.
This same idea — a rule for the rate, stepped forward through time — shows up anywhere something
changes continuously:
Epidemiologyhow fast a disease spreads through a population
Chemistryhow fast reactants turn into products, like PET into TPA + EG
Medicinehow a drug concentration falls in the bloodstream over time
Ecologyhow predator and prey populations rise and fall together
Physicshow a falling object's speed changes due to gravity and drag
Engineeringhow a bridge or building responds to vibration over time