Category Archives: Differential Equations

Why Differential Equations?

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Differential equations are equations that include derivatives. Their solution is not a number, but rather a function which along with its derivative(s) satisfies the equation. That is, when the function and its derivative(s) are substituted into the differential equation the result is true (an identity). You may check your solution by substituting into the differential equation.

Differential equations are used in all areas of math, science, economics, engineering, and anywhere math is used. Derivatives model the change in something. Change is often easier to model (measure and write equations for) than the function that is changing. By solving the differential equation, you find the equation that describes the situation.

If it were only that easy. Differential equations are notoriously difficult to solve. In this, your first look at them, you will study the basics and only one of the many, many methods of solution. This is just to give you a hint of what differential equations are about.

Solution involves finding antiderivatives that include a constant of integration. The solution with an unevaluated constant is called the general solution. The solution could go through any point in the plane depending on the value of the constant of integration.  

To evaluate this constant, you must know a point on the solution function. This is called an initial condition, an initial point, or a boundary condition. Once the constant is evaluated, the result is called the particular solution.

A slope field is a technique for looking at all the solutions and seeing properties of the solutions. A slope field is a series of short segments regularly spaced over the plane that have the slope indicated by the differential equation. The segments are tangent to the solution curve through the points where they are drawn. You may start at any point (the initial condition point) and sketch an approximate solution by following the slope field segments. Doing so gives you an idea of a particular solution.

You will look at exponential functions as an example of an application of a differential equation.

BC students will also learn a numerical approximation technique called Euler’s Method. This is based on the linear approximation idea repeated several times. They will also look another model for the
Logistic equation.

Course and Exam Description Unit 7

Differential Equations (Type 6)

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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
Look at the scoring standards to learn how the solution of the differential equation is scored, and therefore, how students should present their answer. This is usually the one free-response answer with the most points riding on it. Starting in 2016 the scoring has changed slightly. The five points are now distributed this way:
  • 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.
In the past, the domain of the solution was often included on the scoring standard, but unless it was specifically asked for in the question students did not need to include it. However, the 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

Unit 7 – Differential Equations

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Unit 7 is an introduction to the initial ideas and easy techniques related to differential equations . (CED – 2019 p. 129 – 142 ). These topics account for about 6 – 12% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 7.1 – 7.9

Topic 7.1 Modeling Situations with Differential Equations Relating a functions and its derivatives.

Topic 7.2 Verifying Solutions for Differential Equations A proposed solution of a differential equation can be checked by substituting the function and its derivative(s) into the original differential equation. There may be an infinite number of general solutions (solutions with one or more constants).

Topic 7.3 Sketching Slope Fields Slope fields are a graphical representation of a differential equation and provide information about the behavior of the solutions.

Topic 7.4 Reasoning Using Slope Fields 

Topic 7.5 Approximating Solutions Using Euler’s method (BC ONLY) A numerical approach to approximating solutions of a differential equation.

Topic 7.6 Finding General Solutions Using Separation of Variable Since this unit is only an introduction to differential equations, the method of separation of variable is the only solution method tested on the AB and BC exams.

Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables An initial condition (i.e. a point on the particular solution) allows you to evaluate the constant in the general solution and find the one solution that contains the initial condition. Also, if \displaystyle \frac{{dy}}{{dx}}=f\left( x \right) has the initial condition\displaystyle \left( {a,F(a))} \right), then the solution is\displaystyle F\left( x \right)=F\left( a \right)+\int_{a}^{x}{{f\left( x \right)dx}}. Solution may also be subject to domain restrictions. 

Topic 7.8 Exponential Models with Differential Equations Applications include linear motion and exponential growth and decay. The growth and decay model is \displaystyle \frac{{dy}}{{dt}}=kt with the initial condition \displaystyle \left( {0,y\left( 0 \right)} \right) has the solution \displaystyle y=y\left( 0 \right){{e}^{{kt}}}

Topic 7.9 Logistic Models with Differential Equations (BC ONLY) The model of logistic growth, \displaystyle \frac{{dy}}{{dx}}=ky\left( {a-y} \right), can be solved by separating the variables and using partial fraction decomposition. This has never been tested (probably because solving requires a large amount of complicated algebra). Students are expected to know how to interpret the properties of the solution directly from the differential equation (asymptotes, carrying capacity, point where changing the fastest, etc.) and discuss what they mean in context without actually solving the equation.

Timing

The suggested time for Unit 7 is  8 – 9 classes for AB and 9 – 10 for BC of 40 – 50-minute class periods, this includes time for testing etc.

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

Differential Equations  A summary of the terms and techniques of differential equation 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. 

Posts on BC Only Topics

Euler’s Method

Euler’s Method for Making Money

The Logistic Equation 

Logistic Growth – Real and Simulated

Adapting 2021 AB 6

Adapting 2021 BC 5

Adapting 2021 BC 5

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Eight of nine. We continue our study of the 2021 free-response questions. We will look at ways to adapt, expand, and explore this question to help students better understand it and look at other questions that can be asked based on a similar stem.

2021 BC 5

This is a Differential Equation (Type 6) with a Sequence and Series (Type 10) question included. It contains topics from Units 7 and 10 of the current Course and Exam Description (CED). It is not unusual for AP Calculus exam question to include several of the types in my classification and from several of the units from the CED (here units 7 and 10). In addition, the usual solving an initial value problem and a Euler’s Method approximation are included.

The stem for 2021 BC 5 is:

Part (a): Students were asked to write the second-degree Taylor polynomial for the function centered at x = 1 and then use it to approximate f(2). Students should stop after substituting 2 into their polynomial; no arithmetic or simplification is required, and a simplifying mistake will lose a point.

Discussion and ideas for adapting this question:

  • Ask students to find an expression for the second derivative (implicit differentiation).
  • Verify that \displaystyle {f}''\left( 1 \right)=4
  • Ask students to find the third-degree polynomial and use it to approximate f(2)

Part (b): Students were required to approximate f(2) using Euler’s method with two steps of equal size.

Discussion and ideas for adapting this question:

  • After you solve the equation in part (c), ask students to compare the approximations from parts (a) and (b) with the exact value. Neither approximation is very close to the exact value. Discuss why this is so. Consider the slope of the graph near x = 2.
  • Find a more accurate approximation using 3, 4, or more smaller steps. There are graphing calculator programs that will do the arithmetic. Do not hesitate to use them. Students have already shown they know how to do a Euler’s Method approximation; the point is to understand the situation.

Part (c): Finding the solution of the differential equation by separating the variables is expected in this kind of question. The added twist is that the method of integration by parts is necessary to find one of the antiderivatives.

 Discussion and ideas for adapting this question:

  • Be sure not to skip over removing the absolute value signs. The most efficient way is to realize that at (and near) the initial condition y > 0, so |y| = y. What do you do if y < 0?
  • There is not much you can do differently here. One thing is to change the initial condition. Try a negative value such as f(1) = –4.
  • As suggested in (b), compare, and discuss the approximations with the exact value.

Next week we will conclude this series of posts with a look at 2021 BC 6.

I would be happy to hear your ideas for other ways to use this question. Please use the reply box below to share your ideas.

Adapting 2021 AB 6

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Six of nine. Continuing the current series of posts, this post looks at the AB Calculus 2021 exam question AB 6. Like most of the AP Exam questions, there is a lot more you can ask based on the stem of this question and a lot of other calculus you can discuss. This series of post offers suggestions as to how to adapt, expand, and use this question to help your students dig deeper and learn more.

2021 AB 6

This is a standard Differential Equation (Type 6) question and contains topics mainly from Unit 7 (Differential equations) and a little from Unit 3 (implicit differentiation) of the current Course and Exam Description. A differential equation with an initial condition is given in a context. The main part is the solution of the initial value question with three short other questions included.

The stem for 2021 AB 6 is:

Part (a): A slope field in the first quadrant with no scale on either axis is given. Students are asked to sketch the solution curve starting at the initial condition, the point (0, 0).  (I prefer this kind of slope field question to those where students are given a few points and asked to sketch the slope field through them. No one draws slope field by hand; slope field drawn by computers are used to study the approximate shape of the solution and determine its interesting properties as is done here and in part (b)). When drawing slope fields, the sketch should extend to from one border to another and contain the initial condition point.

Discussion and ideas for adapting this question:

  • Have student sketch solution through one or more different points. Copy the slope field and add the initial condition point somewhere else.
  • Add an initial point above the horizontal asymptote.
  • Compare and contrast the solutions drawn through several points.
  • Ask what the horizontal segments (at y = 12) tells you in the context of the problem.

Part (b):  Students are given the limit at infinity for the as yet unknow solution and asked to interpret it in the context of the problem including units of measure.

Discussion and ideas for adapting this question:

  • Discuss why this is so.
  • Discuss how to determine the units of the function from the given information.
  • Discuss how to determine the units of the derivative from the given information.
  • Discuss how to determine the units of the derivative from the units of the function.
  • Discuss how to determine the units of the function from the units of the derivative.
  • Discuss whether the interpretation of the limit makes sense in the context of the question.

Part (c): Students are asked to solve the initial value question using the method of separation of variables.

Discussion and ideas for adapting this question:

  • Since separation of variables is the only method for solving a differential equation that students are responsible for knowing, there is not much you can do to adapt or change this question.
  • The initial condition may be substituted immediately after the integration is done the “+ C” is attached, or it may be done later after the expression is solved for y. Show students both method and discuss which is more efficient and which makes more sense to them.
  • Removing the absolute value signs is another place that may confuse students. While some textbooks suggest using a “ ± “ sign and deciding sign which to use later, the better way is to decide as soon as possible. Ask yourself is the expression enclose by the absolute value signs positive or negative near (at) the initial value. If positive, then the absolute value is replaced by the same expression (as in this question); if negative, then the expression is replaced by its opposite. Then complete the question from there.

Part (d): This part needs careful reading. Students are asked, for a slightly different differential equation, if the rate of change in the amount of medicine is increasing or decreasing at a given time. Therefore, students must find the rate of change of the rate of change (the given derivative): the derivative of the derivative (i.e., the second derivative of the function). This requires implicit differentiation of the derivative using the quotient rule.

Discussion and ideas for adapting this question:

  • The second derivative has the first derivative as one of its factors. Students may (automatically) substitute the first derivative before simplifying or evaluating. This correct, but unnecessarily long. Show the students how to find and substitute the value of the first derivative along with the other numbers.
  • Do as little arithmetic as possible. You need only determine if the second derivative is positive or negative.
  • Discuss the meaning of the answer in the context of the problem.

Next week 2021 BC 2

I would be happy to hear your ideas for other ways to use this question. Please use the reply box below to share your ideas.

Differential Equations – Unit 7

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Unit 7 is an introduction to the initial ideas and easy techniques related to differential equations . (CED – 2019 p. 129 – 142 ). These topics account for about 6 – 12% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 7.1 – 7.9

Topic 7.1 Modeling Situations with Differential Equations Relating a functions and its derivatives.

Topic 7.2 Verifying Solutions for Differential Equations A proposed solution of a differential equation can be checked by substituting the function and its derivative(s) into the original differential equation. There may be an infinite number of general solutions (solutions with one or more constants).

Topic 7.3 Sketching Slope Fields Slope fields are a graphical representation of a differential equation and provide information about the behavior of the solutions.

Topic 7.4 Reasoning Using Slope Fields 

Topic 7.5 Approximating Solutions Using Euler’s method (BC ONLY) A numerical approach to approximating solutions of a differential equation.

Topic 7.6 Finding General Solutions Using Separation of Variable Since this unit is only an introduction to differential equations, the method of separation of variable is the only solution method tested on the AB and BC exams.

Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables An initial condition (i.e. a point on the particular solution) allows you to evaluate the constant in the general solution and find the one solution that contains the initial condition. Also, if \displaystyle \frac{{dy}}{{dx}}=f\left( x \right) has the initial condition \displaystyle \left( {a,F\left( a \right)} \right), then the solution is \displaystyle F\left( x \right)=F\left( a \right)+\int_{a}^{x}{{f\left( x \right)dx}}. Solution may also be subject to domain restrictions. 

Topic 7.8 Exponential Models with Differential Equations Applications include linear motion and exponential growth and decay. The growth and decay model is \displaystyle \frac{{dy}}{{dt}}=ky with the initial condition\displaystyle (0,{{y}_{0}}) has the solution \displaystyle y={{y}_{0}}{{e}^{{kt}}}.

Topic 7.9 Logistic Models with Differential Equations (BC ONLY) The model of logistic growth, \displaystyle \frac{{dy}}{{dt}}=ky\left( {a-y} \right), can be solved by separating the variables and using partial fraction decomposition. This has never been tested (probably because solving requires a large amount of complicated algebra). Students are expected to know how to interpret the properties of the solution directly from the differential equation (asymptotes, carrying capacity, point where changing the fastest, etc.) and discuss what they mean in context without actually solving the equation.

Timing

The suggested time for Unit 7 is  8 – 9 classes for AB and 9 – 10 for BC of 40 – 50-minute class periods, this includes time for testing etc.

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

Differential Equations  A summary of the terms and techniques of differential equation 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 EquationsAn 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. 

Posts on BC Only Topics

Euler’s Method

Euler’s Method for Making Money

The Logistic Equation 

Logistic Growth – Real and Simulated

Differential Equations (Type 6)

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AP  Questions Type 6: Differential Equations

Differential equations are tested almost every year. The actual solving of the differential equation is usually the main part of the problem, but it is 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. Large parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam. 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).
  • NEW Determine the domain restrictions on the solution of a differential equation. See this post for more on this. 
  • 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

Look at the scoring standards to learn how the solution of the differential equation is scored, and therefore, how students should present their answer. This is usually the one free-response answer with the most points riding on it. Starting in 2016 the scoring has changed slightly. The five points are now distributed this way:

  • 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.” and
  • 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..

In the past, the domain of the solution is often included on the scoring standard, but unless it is specifically asked for in the question students do not need to include it. However, the 9 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 2019 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

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 equation 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