Why do populations explode? Why do radioactive materials decay? Why does your coffee cool quickly at first, then slower? In each case, the rate of change is proportional to the current amount. This single principle, expressed as $\frac{dy}{dt} = ky$, governs an astonishing range of natural phenomena.
This differential equation is the foundation for modeling growth and decay. Master it and you can predict populations, date ancient artifacts, and understand how investments compound.
Can you answer these? If not, review the linked prerequisite first.
| Property | Value |
|---|---|
| Concept | Differential Equations |
| Chapter | 6, Section 5 |
| Difficulty | Beginner |
| Time | ~15 minutes |
The law of natural growth (or decay) states:
$$\boxed{\frac{dy}{dt} = ky}$$
where:
Interpretation:
Theorem: The only solutions of $\frac{dy}{dt} = ky$ are:
$$\boxed{y(t) = y(0)e^{kt} = Ce^{kt}}$$
where $C = y(0)$ is the initial value.
Why is $y = Ce^{kt}$ the solution? Differentiate:
$$\frac{dy}{dt} = \frac{d}{dt}[Ce^{kt}] = C \cdot ke^{kt} = k(Ce^{kt}) = ky \quad \checkmark$$
| Context | Name for $k$ | Units |
|---|---|---|
| Population | Relative growth rate | per year, per hour |
| Radioactive decay | Decay constant | per year, per second |
| Finance | Interest rate | per year |
| Temperature | Cooling constant | per minute |
The quantity $\frac{dy/dt}{y} = k$ is called the relative growth rate, the rate of change per unit of current value.
Example: If $k = 0.03$ per year, the quantity grows at 3% per year.
For an investment with principal $A_0$ and interest rate $r$ compounded continuously:
$$A(t) = A_0 e^{rt}$$
This follows from $\frac{dA}{dt} = rA$ (the rate of increase equals the interest rate times the current amount).
| Mistake | Correction |
|---|---|
| Writing $y(t) = e^{kt}$ without initial value | Always write $y(t) = y_0 e^{kt}$ or $y(t) = Ce^{kt}$ |
| Confusing $k$ with the percentage change | $k = 0.05$ means 5% relative rate, but actual growth after 1 unit is $e^{0.05} - 1 \approx 5.13\%$ |
| Thinking decay reaches zero | $e^{kt} \to 0$ as $t \to \infty$, but never equals 0 |
| Using $\ln$ incorrectly: $\ln(a/b) \neq \ln a / \ln b$ | Correct: $\ln(a/b) = \ln a - \ln b$ |
| Mixing time units | If $k$ is per hour, $t$ must be in hours |
Problem: A quantity satisfies $\frac{dy}{dt} = 0.05y$ with $y(0) = 200$. Find $y(t)$ and determine when the quantity doubles.
Solution:
Step 1: Apply the general solution. $$y(t) = y(0)e^{kt} = 200e^{0.05t}$$
Step 2: Find the doubling time by setting $y(t) = 400$. $$200e^{0.05t} = 400$$ $$e^{0.05t} = 2$$ $$0.05t = \ln 2$$ $$t = \frac{\ln 2}{0.05} = 20\ln 2 \approx 13.86$$
The quantity doubles in approximately 13.86 time units.
Verify that $y(t) = 5e^{-2t}$ is a solution to the differential equation $\frac{dy}{dt} = -2y$.
Solve the initial value problem: $\frac{dy}{dt} = 3y$, $y(0) = 7$.
A quantity grows exponentially. At $t = 0$ it is 50, and at $t = 4$ it is 150. Find the growth constant $k$ and write the formula for $y(t)$.
An investment of $8000 earns interest compounded continuously.
Prove that if $y(t)$ is any solution to $\frac{dy}{dt} = ky$ with $y(0) = C$, then $y(t) = Ce^{kt}$.
Hint: Consider the function $g(t) = y(t)e^{-kt}$ and show that $g'(t) = 0$.
Question 1: If a quantity satisfies $\frac{dy}{dt} = -0.1y$, is the quantity growing or decaying? What is the relative rate of change?
Decaying, because $k = -0.1 < 0$. The relative rate of change is $-0.1$ or $-10\%$ per unit time.
Question 2: Two quantities both satisfy exponential growth equations. Quantity A has $k = 0.02$ and quantity B has $k = 0.05$. If both start at the same value, which grows faster? By what factor will B exceed A after 10 time units?
B grows faster (larger $k$). After 10 units:
B exceeds A by a factor of about 1.35.
Question 3: The doubling time for quantity X is 5 years. The doubling time for quantity Y is 10 years. What is the ratio of their growth constants?
Doubling time $T_d$ satisfies $e^{kT_d} = 2$, so $k = \frac{\ln 2}{T_d}$.
Ratio: $\frac{k_X}{k_Y} = \frac{10}{5} = 2$
The "Interest on Interest" Analogy:
Think of exponential growth like compound interest that compounds every instant. Each tiny moment, you earn interest not just on your principal, but on all the interest you've already earned. The more you have, the more you gain; this feedback loop creates the exponential curve.
| If you're struggling with... | Review this |
|---|---|
| The derivative of $e^{kt}$ | Derivative of Exponential Functions |
| Solving equations with $\ln$ | Natural Logarithm |
| What a differential equation is | This is a first-order ODE; see Introduction to Differential Equations |
Looking back:
Looking ahead:
Real-world connections:
| Previous | Up | Next |
|---|---|---|
| §6.4 Skills | Chapter 6 Skills | Population Growth |
Last updated: 2026-01-22