# Unit 7 – Differential Equations

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

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.

# Differential Equations – Unit 7

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

# 2019 CED Unit 7: Differential Equations

Applications (Unit 8) seem to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) present differential equations first probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration before Unit 7: Differential Equations. Therefore, for those who want to present unit 8 first, I will post unit 8 next week on December 3, 2019. That way you’ll have both for reference and can choose the order you think will work best for your students.

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 function and its derivatives.

Topic 7.2 Verifying Solutions for Differential Equations A proposed solution to 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 about 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.

### Posts on BC Only Topics

Euler’s Method

Euler’s Method for Making Money

The Logistic Equation

Logistic Growth – Real and Simulated

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series