Topic 7.1 Notes – Modeling Situations with Differential Equations

What a Differential Equation Is in a Modeling Context

A differential equation is an equation that relates a function and one or more of its derivatives.

In AP Calculus AB, that usually looks like:

$$\frac{dy}{dx} = \text{expression involving } x \text{ and/or } y$$

So you’re connecting:

• A quantity (like $y$, $P$, $V$, etc.)
• Its rate of change (a derivative)
• The independent variable (often time $t$)

If you see “rate of change,” your brain should immediately think derivative.

In modeling, the derivative isn’t just a slope from a graph. It represents something real:

• population growth → $\frac{dP}{dt}$
• water leaking from a tank → $\frac{dV}{dt}$
• temperature changing → $\frac{dT}{dt}$

Differential equations describe how something changes, not what it equals.

Translating Verbal Statements into Differential Equations

Most questions in this topic are translation problems. You’re turning English into calculus.

Here’s the structure almost every time:

Step-by-Step Translation

1. Identify the changing quantity
   That becomes the dependent variable.
   “The rate of change of mass” → $\frac{dM}{dt}$
2. Identify the independent variable
   Usually time $t$, but sometimes $x$, distance, etc.
3. Spot proportional language
   • “Proportional to”
   • “Inversely proportional to”
   • “Proportional to the square of”
   • “Proportional to the product of”
4. Introduce a constant of proportionality $k$
   Always include $k$. Never assume it’s 1.

Common Proportional Relationships

Direct Proportionality

“The rate of change of $R$ is proportional to $x$.”

$$\frac{dR}{dt} = kx$$

If it says proportional to the product:

$$\frac{dR}{dt} = kxy$$

If proportional to a power:

$$\frac{dy}{dx} = kx^3$$

Inverse Proportionality

“The rate of change of $S$ is inversely proportional to $x$.”

$$\frac{dS}{dt} = \frac{k}{x}$$

Direct means multiply.
Inverse means divide.

That distinction is one of the most common quiz mistakes.

Proportional to Itself

“The rate of change of $P$ is proportional to $P$.”

$$\frac{dP}{dt} = kP$$

This is the foundation of exponential growth and decay (you’ll solve these later).

Finding the Constant of Proportionality $k$

Often you’re given specific numerical information.

Example:

“The rate of change of temperature $T$ is proportional to the square of time $t$. When $t = 2$, the temperature is increasing at 12 degrees per minute.”

Step 1: Write the general equation

$$\frac{dT}{dt} = kt^2$$

Step 2: Plug in given values

$$12 = k(2^2)$$

$$12 = 4k$$

Step 3: Solve for $k$

$$k = 3$$

Final model:

$$\frac{dT}{dt} = 3t^2$$

Notice we substituted into the derivative, not into $T$. That’s a common place students slip up.

On free-response questions, you’ll usually earn a point for the correct differential equation and another for correctly finding $k$.

Recognizing the Structure of These Problems

Almost every modeling question in this topic follows this pattern:

1. A quantity is changing.
2. The problem describes how its rate depends on something.
3. The relationship is proportional (direct or inverse).
4. You write the differential equation.
5. Sometimes you determine $k$.

If you train yourself to think through those steps in order, these problems become very mechanical. Here’s a quick visual summary of the logic:

Modeling differential equations workflow

That’s the whole process. On exam day, you want this flow in your head so you can move from the words to the equation quickly and confidently.

Multiple-choice questions often test whether you picked the correct structure, especially distinguishing between $\frac{dy}{dt} = ky$ and $\frac{dy}{dt} = \frac{k}{y}$.

Common Mistakes That Cost Points

• Writing $y = kx$ when the problem says “rate of change” (you forgot the derivative).
• Forgetting the constant $k$.
• Mixing up direct and inverse proportionality.
• Using the wrong variable in the derivative (writing $\frac{dS}{dx}$ when the problem says “with respect to time”).

Be precise with notation. The AP graders care about it.