Modeling Situations with Differential Equations

What Differential-Equation Modeling Means

A differential equation describes a relationship involving an unknown function and one or more of its derivatives. In AP Calculus, you are often given a verbal description of how a quantity changes and asked to translate that description into a differential equation.

The key idea is simple: look for a statement about a rate of change. Words such as "increases at a rate," "decreases at a rate," "proportional to," and "depends on" are clues that a derivative belongs in your equation.

For example, if represents the amount of a substance at time , then represents how quickly that amount is changing.

Turning Words into Equations

The most important modeling skill is translating ordinary language into mathematical language. Start by defining the dependent variable and the independent variable before writing the equation.

Common Translation Patterns

"The rate of change of is proportional to ." →
"The population increases at a rate proportional to its current population." →
"The amount increases at a constant rate of ." →
"The temperature changes at a rate proportional to the difference between the temperature and the surrounding temperature." →

Notice that "proportional to" almost always introduces a constant of proportionality. If one quantity is proportional to another, write , where is a constant.

A Reliable Modeling Method

Define the variables. State what the dependent variable represents and what the independent variable represents.
Identify the rate. Look for language describing how the quantity changes. This usually gives you a derivative.
Translate the relationship. Convert phrases such as "proportional to," "constant rate," or "difference from equilibrium" into mathematical expressions.
Include the constant of proportionality when the problem says one quantity is proportional to another.
Use initial conditions separately. An initial value such as is not part of the differential equation itself; it tells you which particular solution describes the situation.
Check the units and signs. A rate should have units of "quantity per unit time," and the sign should agree with whether the quantity is increasing or decreasing.

Worked Examples

Example 1: Population Growth

A population grows at a rate proportional to the current population. Write a differential equation that models the situation.

Step 1: Define as the population at time .

Step 2: "Grows at a rate" tells us to use .

Step 3: "Proportional to the current population" means the rate is proportional to .

Result:

Example 2: Constant Rate of Change

A tank contains liters of water. Water is being added at a constant rate of liters per minute. Let represent the amount of water after minutes. Write a differential equation and an initial condition.

Step 1: Since the amount is changing with respect to time, the rate is .

Step 2: The water is being added at liters per minute, so .

Step 3: Initially the tank contains liters, so .

Result: The model is together with the initial condition .

Example 3: Newton's Law of Cooling

A cup of coffee is placed in a room maintained at . Let be the temperature of the coffee at time . Suppose the rate at which the coffee's temperature changes is proportional to the difference between its temperature and the room temperature.

Step 1: The rate of temperature change is .

Step 2: The difference between the coffee temperature and the room temperature is .

Step 3: Because the coffee cools when it is hotter than the room, the rate should be negative when . A convenient model is , where .

Result:

Example 4: Logistic Growth

A population grows rapidly when it is small, but its growth slows as the population approaches a carrying capacity . Let be the population at time . A logistic model assumes that the growth rate is proportional both to the current population and to the amount of capacity remaining.

Step 1: The current population is .

Step 2: The amount of capacity remaining is .

Step 3: "Proportional both to" means multiply these factors and introduce a constant .

Result:

This model captures an important idea: growth can be positive while still slowing down. When becomes small, the growth rate becomes small.

Initial Conditions Are Part of the Model

A differential equation usually describes an entire family of possible functions. An initial condition selects the particular function that matches the real-world situation.

For example, the differential equation has many possible solutions. If the problem also tells us that , then we can use that information to determine the particular solution relevant to the situation.

Interpreting the Differential Equation

AP Calculus questions often ask you to interpret a differential equation rather than solve it. Every part of the equation should have a meaning in context.

Suppose a population satisfies . The left side, , is the instantaneous rate at which the population changes. The right side says that this rate depends on the current population.

If is positive, the population is increasing. If is negative, the population is decreasing. If , the population is momentarily neither increasing nor decreasing.

Modeling Inflow and Outflow

Many AP Calculus modeling problems involve a quantity entering and leaving a system. The fundamental principle is:

This is often called an input minus output model. If a tank receives water at rate and loses water at rate , then the amount of water satisfies:

The same structure applies to populations, money, chemicals, medication, pollutants, and many other quantities.

Common Mistakes

Writing the quantity instead of its rate

If a problem says "the amount increases at a rate proportional to the amount," the differential equation must involve , not just . The phrase "rate of change" is the signal that a derivative belongs in the equation.

Forgetting the constant of proportionality

If the rate is proportional to a quantity, write a constant such as . Writing only incorrectly assumes that the constant of proportionality is exactly 1.

Getting the sign wrong

A quantity that is decreasing should have a negative rate. For example, if a population decreases at a rate proportional to its current size, the model should look like with , or equivalently with .

Confusing an initial condition with the differential equation

The equation describes how the quantity changes; the initial condition tells you where the system starts. For example, is a differential equation, while is an initial condition.

Practice Problems

Question 1

A population grows at a rate proportional to its current size. Write a differential equation modeling the population.

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Question 2

A tank contains liters of water initially. Water enters the tank at liters per minute, while water leaves at liters per minute. Write a differential equation and initial condition for the amount of water.

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Question 3

A temperature is surrounded by an environment maintained at . The rate of change of temperature is proportional to the difference between the temperature and the environmental temperature. Write a differential equation using a positive constant .

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Question 4 (Logistic Model)

A population has carrying capacity . Its growth rate is proportional to both its current population and the amount of capacity remaining. Write a logistic differential equation.

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