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Differential equations · 02 · The most important equation · 9 min

Exponential growth & decay

One law rules them all: y′ = ky — change proportional to current amount. Its solution is the exponential function, and it governs money, epidemics, populations, radioactivity, and cooling coffee alike.

Build the intuition

Why proportional growth makes exponentials

More bacteria make more bacteria; more money earns more interest. When the growth rate is a fixed fraction of the current amount, each unit of time multiplies the total by the same factor — and repeated multiplication is exactly what an exponential is. The calculus seals it: the function whose derivative is k times itself is Ce^{kt}, full stop.

y′=ky  ⇒  y=Cekty' = ky \;\Rightarrow\; y = Ce^{kt}

Doubling time and half-life

Exponentials have a signature: a fixed time to double (k > 0) or halve (k < 0), no matter the current size. Money at 7% doubles every ~10 years — $1k→$2k takes as long as $1M→$2M. Carbon-14 halves every 5,730 years, which is the entire mechanism of radiocarbon dating.

tdouble=ln⁡2kt_{double} = \frac{\ln 2}{k}

Why exponentials fool everyone

Early exponential growth looks flat — then suddenly vertical. The lily pond that doubles daily and fills in 30 days was half full on day 29, and 3% full just five days before the end. Pandemics and viral products both exploit this blind spot in human intuition. The equation sees it coming; eyeballs don't.

See it move

InteractiveA field of instructions
0.6
1
The law y′ = 0.6·y plants a tiny slope at every point. Starting from y(0) = 1, the solution y = 1e^{0.6t} simply follows the grain — compounding upward (growth).

Positive k: the field funnels every start into explosion. Negative k: everything decays to zero. One dial, both destinies.

A worked example

Date the ancient bone

  1. A bone retains 25% of its original carbon-14. Carbon-14's half-life is 5,730 years.

  2. 25% is two halvings:

    100%→50%→25%100\% \to 50\% \to 25\%

  3. Age ≈ 2 × 5,730 = 11,460 years. Archaeology's clock is the decay law y′ = −ky, read backwards.

Out in the world

Drug dosing schedules

Medications clear the bloodstream exponentially — each drug has a half-life. “Take every 8 hours” is computed so that decay between doses keeps levels inside the effective-but-safe window. Your prescription label is a solved differential equation.

Common confusion, cleared

“Exponential just means “really fast.””

It means proportional-to-itself — which starts slow and finishes absurd. Early on, exponentials lose races to straight lines; given time, they beat everything.

“Doubling time shrinks as the amount grows.”

It's constant — that's the fingerprint. If doubling is speeding up or slowing down, the process isn't purely exponential (see the next lesson's limits to growth).

Check yourself

PracticeQuick check

  1. Which situation obeys y′ = ky?

  2. A culture doubles every 3 hours, starting at 1,000. After 9 hours…

Recap

  • y′ = ky ⇒ y = Ce^{kt}: proportional change makes exponentials.
  • Fixed doubling time (ln 2 / k) or half-life is the signature.
  • Exponentials look flat early, vertical late — trust the math over the eye.