The Exponential Growth and Decay Model
One assumption generates the most important differential equation in applied mathematics: a quantity changes at a rate proportional to its current size. Money earns interest on its balance; populations breed in proportion to their numbers; radioactive atoms decay each with the same probability per second. All of it is the single equation
dy/dt = k·y, with solution y(t) = y₀·e^(kt)
Positive k means growth, negative k decay, and |k| sets the tempo. The solution follows in two lines by separation of variables: dy/y = k dt integrates to ln|y| = kt + C, and exponentiating with y(0) = y₀ gives the formula. Everything about compound growth, doubling times, half-lives, rule-of-72 folklore, carbon dating, is unpacked from this one exponential.
How to Use This Calculator
Enter the initial amount y₀ and the rate constant k (per unit time; use a negative k for decay). The calculator writes the particular solution, reports the doubling time (growth) or half-life (decay), and, given a time t, evaluates y(t), plotting the curve with the point marked. The chips cover the canon: continuous 5% interest, carbon-14's famous 5730-year half-life at t = one half-life (returning exactly half the sample), a 3-hour-half-life medication after two half-lives, and k = ln 2 ≈ 0.6931, the rate that doubles every time unit.
Worked Example: Carbon Dating
Carbon-14 decays with k = −0.000121 per year. A sample starting at 100 units, evaluated at t = 5730 years:
y(5730) = 100·e^(−0.000121 × 5730) = 100·e^(−0.6933) ≈ 50.0
Half the carbon-14 remains, because 5730 years is the half-life, t½ = ln 2/|k| = 0.6931/0.000121 ≈ 5728 years. Archaeologists run the logic backward: measure the surviving fraction p, solve p = e^(kt) for t = ln(p)/k, and the sample dates itself. A bone retaining 25% of its carbon-14 has waited two half-lives, about 11,460 years.
Half-Life, Doubling Time, and the Meaning of k
The rate constant k is the continuous (instantaneous) percentage rate: k = 0.05 means growing at 5% per unit time at every instant, compounding continuously into slightly more than 5% per whole period (e^0.05 ≈ 1.0513). Two derived clocks are usually more intuitive than k itself: the doubling time t₂ = ln 2/k for growth, and the half-life t½ = ln 2/|k| for decay. Both are constant, a hallmark unique to exponentials: the population doubles every t₂ regardless of its current size, and a radioactive sample halves every t½ regardless of how much remains. When data shows a constant doubling time, the process is exponential; when doubling slows as the quantity grows, you're watching logistic saturation instead.
Common Mistakes to Avoid
- Confusing continuous and per-period rates. "5% annual growth" as discrete compounding is y₀(1.05)ᵗ; as continuous rate it's y₀e^(0.05t). They differ (1.05 vs 1.0513 per year), convert with k = ln(1 + r) when the data is per-period.
- Using half-life as "gone in two half-lives." Halving twice leaves a quarter, not zero, decay never finishes. After n half-lives, (½)ⁿ remains; the 10% mark takes about 3.32 half-lives.
- Fitting an exponential to saturating data. Real populations, epidemics, and markets bend away from exponential once resources bind. The model is the early-phase truth; extrapolating it decades out is the classic forecasting sin.
- Sign slips in k. A decay problem with positive k grows silently. Anchor with a sanity check: does y(t) move in the direction the story says by a plausible amount at small t?
Real-World Applications
Finance runs on this page: continuous compounding, present-value discounting (decay of future money at rate r), and the rule of 72 (72/percentage ≈ doubling time, the mental-math cousin of ln 2/k). Medicine doses by it: first-order drug elimination means plasma concentration decays exponentially, so dosing intervals are set in half-lives, and five half-lives is the standard "effectively cleared" horizon. Nuclear science is it: reactor waste management, medical isotope logistics (technetium-99m's 6-hour half-life dictates same-day hospital delivery), and radiometric dating from carbon-14 to uranium-lead.
Beyond those, atmospheric pressure decays exponentially with altitude, light attenuates exponentially through absorbing media (Beer's law, the basis of pulse oximetry), capacitor charge drains as e^(−t/RC) in every timing circuit, hot objects cool toward ambient exponentially (Newton's law of cooling, forensic time-of-death estimation), and epidemiologists track outbreaks through the doubling time of case counts, exactly the ln 2/k arithmetic this calculator performs.
Frequently Asked Questions
How do I find k from two data points?
From y(t₁) = a and y(t₂) = b: k = ln(b/a)/(t₂ − t₁). One decade of data with y going 1000 → 1500 gives k = ln 1.5/10 ≈ 0.0405. From a stated half-life or doubling time, k = ∓ln 2/t instead.
Why does e appear rather than 2 or 10?
Any base works algebraically, y₀·2^(t/t₂) is the same function, but base e makes the rate constant equal the derivative's proportionality: (e^(kt))′ = k·e^(kt), no stray ln factors. e is the natural currency of "rate proportional to amount"; other bases carry a conversion fee.
What's the difference between exponential and logistic growth?
Logistic growth is exponential with a ceiling: y′ = ky(1 − y/M) grows at rate ≈ k when small, then saturates at carrying capacity M. Every real "exponential" eventually goes logistic; the exponential model is its faithful early chapter.
Is continuously compounded interest actually used?
Yes, it's the standard in quantitative finance (option pricing, yield curves, discount factors all use e^(rt)) precisely because continuous rates add cleanly across time and instruments. Bank retail products quote discrete APY, but the math under the hood is this page's.
How is this related to Newton's law of cooling?
Cooling obeys T′ = −k(T − T_amb): the excess temperature over ambient decays exponentially. Substitute u = T − T_amb and it's exactly y′ = −ku. Many models are exponential decay in disguise, one substitution away.
What does a negative y₀ mean?
The model is linear, so negative amounts propagate fine mathematically (debt compounding is exactly this). The formula y(t) = y₀e^(kt) just scales the whole curve by y₀, sign included, the doubling and halving clocks are unchanged.