Modeling situations with differential equations — AP Calculus AB
1. What Is Modeling with Differential Equations? ★★☆☆☆ ⏱ 3 min
A differential equation is any equation that contains one or more derivatives of an unknown function. Modeling with differential equations is the core process of translating verbal, contextual, or graphical descriptions of relationships between a quantity and its rate of change into a formal mathematical equation. This is always the first step in almost all differential equation problems on the AP exam: you cannot solve a differential equation if you cannot write it correctly from the given description.
According to the AP Calculus AB CED, Unit 7 (Differential Equations) makes up 6-12% of the total exam weight, and this specific topic is the gateway to all other differential equation skills in the unit, including slope fields, separation of variables, and exponential growth/decay. It appears in both multiple-choice and free-response questions, almost always as the opening part of a longer FRQ that leads into solving the differential equation or interpreting its solution.
2. Translating Contextual Rate Descriptions ★★★☆☆ ⏱ 4 min
The core idea of this skill is that any description of how a quantity changes automatically translates to the derivative of that quantity. The most common scenario is a rate of change proportional to the quantity itself, but you will also see rates proportional to a difference between the quantity and a fixed constant, rates proportional to a power of the quantity, and rates equal to a combination of multiple terms.
- Explicitly identify your dependent variable (the quantity that changes) and independent variable (almost always time $t$ in AP problems).
- Translate the phrase "the rate of change of [dependent variable]" directly to the derivative of the dependent variable with respect to the independent variable.
- "Is proportional to X" always means "equals $k$ times X", where $k>0$ is the constant of proportionality by convention.
- Add the correct sign: if the quantity is decreasing, the derivative will have a negative sign, even with $k>0$.
Exam tip: Always translate the problem word-for-word, do not add extra assumptions. If a problem says "the difference between $y$ and 100", write $(y - 100)$, do not swap it to $(100 - y)$ unless the problem explicitly tells you the direction of change.
3. Classifying Differential Equations by Order ★★☆☆☆ ⏱ 3 min
The order of a differential equation is defined as the order of the highest-order derivative that appears in the equation. For example, a first-order differential equation only contains first derivatives of the unknown function, while a second-order differential equation contains a second derivative. AP Calculus AB almost exclusively works with first-order differential equations, but you are expected to correctly identify the order of any DE, a common trick on MCQs.
Order matters because the number of arbitrary constants in the general solution of a DE is equal to the order of the DE, so first-order DEs require one initial condition to find a particular solution, which is the standard AP problem.
Exam tip: When asked for the order, circle every derivative in the DE and write its order next to it, then pick the maximum number. This eliminates the common mistake of confusing order with power.
4. Verifying Solutions to Differential Equations ★★★☆☆ ⏱ 4 min
To confirm a given candidate function is a solution to a differential equation, you substitute the function and all its required derivatives into the DE and check that both sides of the equation are equal for all values of the independent variable in the domain. This is a regularly tested skill on both MCQ and FRQ, often as the second part of a question after setting up the DE.
- Compute all derivatives of the candidate function that appear in the DE (for a first-order DE, you only need the first derivative).
- Substitute the candidate function $y$ and its derivatives into the left-hand side and right-hand side of the DE.
- Simplify both sides. If they are identical, the candidate is a solution; if not, it is not.
Exam tip: Double-check your differentiation and arithmetic when verifying solutions: small sign errors can lead you to incorrectly reject a valid solution or accept an invalid one.
Common Pitfalls
Why: Students remember "proportional" but forget to account for the direction of change described in the problem.
Why: Students confuse the order of the derivative (how many times you differentiate) with the power the derivative is raised to.
Why: Students rush and forget that a DE requires the derivative to be substituted.
Why: Students assume $y$ is always approaching 100 so swap the order, but the problem explicitly states the order of the difference.
Why: Students know proportional means multiply, but forget that proportionality requires a constant of proportionality, which is not the variable itself.