Exponential Growth and Decay

The Most Important Equation

There is one differential equation that appears more often than any other in science, engineering, economics, and everyday life:

In words: the rate of change of something is proportional to how much of it there is.

This deceptively simple statement captures an enormous range of phenomena. Bacteria multiply faster when there are more of them. Radioactive atoms decay at a rate proportional to how many remain. Money in a savings account grows faster when there is more principal to earn interest on.

The constant $k$ determines the behavior. When $k > 0$, the quantity grows. When $k < 0$, it decays. The magnitude of $k$ controls how quickly this happens.

Deriving the Solution

This equation is separable: the right side $ky$ can be written as $g(t) \cdot h(y)$ where $g(t) = k$ (a constant function of $t$) and $h(y) = y$. Let us apply separation of variables to derive the solution.

Starting with $\frac{dy}{dt} = ky$, we separate variables:

Integrating both sides:

Exponentiating both sides:

Since $e^C$ is a positive constant and $y$ could be positive or negative, we write $y = \pm e^C \cdot e^{kt}$. Combining $\pm e^C$ into a single constant (which can now be positive or negative) and evaluating at $t = 0$:

Note: We divided by $y$, assuming $y \neq 0$. The constant function $y(t) = 0$ is also a solution (check: $0' = 0 = k \cdot 0$). For nonzero initial conditions, the uniqueness theorem guarantees solutions never cross $y = 0$, so our derivation remains valid.

This is the exponential function. Every solution to $\frac{dy}{dt} = ky$ has this form, differing only in the initial value $y_0$ and the constant $k$.

Growth vs. Decay

The sign of $k$ completely determines the qualitative behavior.

When $k > 0$, the exponential climbs ever upward. The larger $k$ is, the steeper the climb. When $k < 0$, the curve descends toward zero but never quite reaches it. At $k = 0$, nothing changes at all: the solution is simply the constant $y = y_0$.

Notice how the curve changes character as $k$ passes through zero. This transition from growth to decay is a key qualitative feature of exponential processes.

Population Growth

One of the oldest applications of exponential growth is population modeling. If each individual in a population has some probability of reproducing per unit time, and the population is large enough to ignore random fluctuations, then the population grows exponentially.

Consider bacteria in ideal conditions. E. coli can divide approximately every 20 minutes when given unlimited nutrients. Starting with a single bacterium, after one hour you have 8. After two hours, 64. After 24 hours, if growth continued unchecked, you would have over $4 \times 10^{21}$ bacteria, weighing thousands of tons.

Of course, exponential growth cannot continue forever. Eventually, bacteria exhaust their nutrients, accumulate waste products, or run out of space. Real populations follow a logistic curve, where growth slows as the population approaches a carrying capacity. But in the early stages, when resources are plentiful relative to the population, exponential growth is an excellent approximation.

This is why exponential growth is sometimes called Malthusian growth, after Thomas Malthus, who observed that populations tend to grow exponentially while food production grows linearly, leading to inevitable crises.

Radioactive Decay

Perhaps the most precise application of exponential decay occurs in radioactive materials. Each atom of a radioactive isotope has a fixed probability of decaying per unit time, independent of what other atoms are doing. This leads directly to $\frac{dN}{dt} = -\lambda N$, where $\lambda$ is the decay constant (positive, but we write $-\lambda$ to emphasize that $N$ is decreasing).

The solution is $N(t) = N_0 e^{-\lambda t}$, where $N_0$ is the initial number of atoms.

A key concept in radioactive decay is the half-life: the time it takes for half the atoms to decay. We find it by solving $N_0 e^{-\lambda t_{1/2}} = \frac{N_0}{2}$:

Different isotopes have vastly different half-lives. Carbon-14, used for dating organic materials, has a half-life of 5,730 years. Uranium-238, used for dating ancient rocks, has a half-life of 4.5 billion years. Polonium-214 has a half-life of 164 microseconds.

The beauty of half-life is that it is independent of how much material you start with. Whether you have one gram or one kilogram, half will be gone after one half-life. Half of what remains will be gone after another half-life. And so on.

Newton's Law of Cooling

When a hot object is placed in a cooler environment, it loses heat. Newton observed that the rate of cooling is approximately proportional to the temperature difference between the object and its surroundings:

Here $T$ is the object's temperature, $T_{\text{ambient}}$ is the surrounding temperature, and $k > 0$ is a constant that depends on the object's properties and its environment.

This is not quite in the form $\frac{dy}{dt} = ky$, but a simple substitution fixes that. Let $u = T - T_{\text{ambient}}$, the temperature excess above ambient. Then $\frac{du}{dt} = \frac{dT}{dt}$ (since $T_{\text{ambient}}$ is constant), and our equation becomes:

The solution is $u(t) = u_0 e^{-kt}$, or in terms of temperature:

The temperature approaches room temperature exponentially. It never quite reaches it, in theory, but gets arbitrarily close. In practice, once the temperature difference becomes small enough, other effects dominate and our model breaks down.

This is why your coffee cools quickly at first, then seems to take forever to reach room temperature. The rate of cooling depends on the temperature difference, and as that difference shrinks, so does the cooling rate.

Compound Interest

Money grows exponentially when interest is compounded. If you have principal $P$ earning annual interest rate $r$, compounded $n$ times per year, after $t$ years you have:

What happens as we compound more and more frequently? Taking the limit as $n \to \infty$:

This is continuous compounding, and it corresponds exactly to the solution of $\frac{dA}{dt} = rA$.

The difference between compounding annually and continuously is not dramatic for typical interest rates and time horizons. But the mathematical connection is profound: the exponential function $e^x$ arises naturally as the limit of compound growth.

For continuous compounding, we can also define a doubling time: the time for an investment to double. Setting $e^{rt_2} = 2$:

A useful approximation is the Rule of 72: divide 72 by the interest rate (as a percentage) to get the approximate doubling time in years. At 6% interest, money doubles in about 12 years.

The Ubiquity of Exponentials

Why does $\frac{dy}{dt} = ky$ appear everywhere? Because proportionality between a quantity and its rate of change is the simplest possible feedback relationship.

When growth reinforces itself, positive feedback creates exponentials. Each new bacterium can divide to create more. Each new dollar earns interest that creates more dollars. Each informed person can spread a rumor to more people.

When decay is self-limiting, negative feedback creates exponentials too. The hotter an object is above its surroundings, the faster it loses heat. The more radioactive atoms remain, the more decays per second. The more of a drug in your bloodstream, the faster your kidneys filter it out.

This equation is the first and simplest in a hierarchy. When growth depends not just on current population but also on available resources, we get the logistic equation. When two populations interact, we get predator-prey systems. But all of these build on the foundation of exponential behavior.

Key Takeaways

• The equation $\frac{dy}{dt} = ky$ states that the rate of change is proportional to the current value
• The solution is always $y(t) = y_0 e^{kt}$, where $y_0$ is the initial value
• When $k > 0$, the quantity grows exponentially; when $k < 0$, it decays
• Half-life is $t_{1/2} = \frac{\ln 2}{|k|}$; doubling time is $t_2 = \frac{\ln 2}{k}$
• Applications include population growth, radioactive decay, cooling, and compound interest
• Exponential growth cannot continue forever in real systems, but is an excellent short-term approximation