**AP Questions ****Type 6****: Differential Equations**

Differential equations are tested in the free-response section of the AP exams almost every year. The actual solving of the differential equation is usually the main part of the problem accompanied by a related question such as a slope field or a tangent line approximation. BC students may also be asked to approximate using Euler’s Method. Several parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam. This topic may also appear in the multiple-choice sections of the exams.
**What students should be able to do**

- Find the
*general solution*of a differential equation using the method of separation of variables (this is the*only*method tested). - Find a
*particular solution*using the initial condition to evaluate the constant of integration – initial value problem (IVP). - Determine the domain restrictions on the solution of a differential equation. See this post for more on the domain of a differential equation.
- Understand that proposed solution of a differential equation is a function (not a number) and if it and its derivative are substituted into the given differential equation the resulting equation is true. This may be part of doing the problem even if solving the differential equation is not required (see 2002 BC 5 – parts a, b and d are suitable for AB)
- Growth-decay problems.
- Draw a slope field by hand.
- Sketch a
*particular solution*on a given slope field. - Interpret a slope field.
- Multiple-choice: Given a differential equation, identify is slope field.
- Multiple-choice: Given a slope field identify its differential equation.
- Use the given derivative to analyze a function such as finding extreme values
- For BC only: Use Euler’s Method to approximate a solution.
- For BC only: use the method of partial fractions to find the antiderivative after separating the variables.
- For BC only: understand the logistic growth model, its asymptotes, meaning, etc. The exams so far, have never asked students to actually solve a logistic equation IVP

- one point for separating the variables
- one point each for finding the antiderivatives
- one point for including the constant of integration and using the initial condition – that is, for writing “+ C” on the paper with one of the antiderivatives and substituting the initial condition; finding the value of
*C*is included in the “answer point.” (In the older exams one point was earned for writing the +*C*and another point for using the initial condition.) - one point for solving for
*y*: the “answer point”, for the correct answer. This point includes all the algebra and arithmetic in the problem including solving for*C*.

**CED.**lists “EK 3.5A3 Solutions to differential equations may be subject to domain restrictions.” Perhaps this will be asked in the future. For more on domain restrictions with examples see this post. Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find. For some previous posts on differential equations see January 5, 2015, and for post on related subjects see November 26, 2012, January 21, 2013, February 1, 6, 2013

**The Differential Equation question covers topics in Unit 7 of the CED.**

**Free-response examples:**

- 2019 There was no DE question in the free-response. You may assume the topic was tested in the multiple-choice sections.
- 2017 AB4/BC4,
- 2016 AB 4, BC 4, (different questions)
- 2015 AB4/BC4,
- 2013 BC 5
- and a favorite Good Question 2 and Good Question 2 Continued
- 2021 AB 6, BC 5 (b), (c)
- 2022 AB5 – sketch solution on slope field, tangent line approximation, solve separable equation.
- 2023 AB 3 / BC 3 – sketch solution on slope field, tangent line approximation, solve separable equation

**Multiple-choice examples from non-secure exams:**

- 2012 AB 23, 25
- 2012 BC: 12, 14, 16, 23

**Previous posts on these topics for both AB and BC include:**

Differential Equations A summary of the terms and techniques of differential equations and the method of separation of variables
Domain of a Differential Equation – On domain restrictions.
Accumulation and Differential Equations
Slope Fields
An Exploration in Differential Equations An exploration illustrating many of the ideas of differential equations. The exploration is here in PDF form and the solution is here. The ideas include: finding the general solution of the differential equation by separating the variables, checking the solution by substitution, using a graphing utility to explore the solutions for all values of the constant of integration, finding the solutions’ horizontal and vertical asymptotes, finding several particular solutions, finding the domains of the particular solutions, finding the extreme value of all solutions in terms of *C,*finding the second derivative (implicit differentiation), considering concavity, and investigating a special case or two.

**Previous**** Posts on BC Only Topics**

Euler’s Method
Euler’s Method for Making Money
The Logistic Equation
Logistic Growth – Real and Simulated
Revised 2/20/2021, March 29, May 14, 2022, June 6, 2023

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