 are at the heart of mathematical modeling for all the sciences: physical, social, financial, et cetera.

 is the use of math to describe physical situations. Essentially, math functions as a language. While some sciences prefer to put emphasis on traditional language to describe phenomena (we use words to describe explain biological happenings, for example) other sciences, like physics, rely on mathematical modeling to express ideas and concepts. 

Mathematical modeling is useful because it often yields logical derivations that could not have been predicted in the first place. For example, the mathematical modeling of the behavior of particles directly led to the conclusion that subatomic particles can travel through a barrier of infinite energy, a deduction that would seem impossible through a different lens

Mathematical modeling also helps scientists confirm logical conclusions. Predictable actions like the behavior of a swinging pendulum with significant air friction

 are supported and confirmed by mathematical modeling. 

Differential equations serve as scaffolding for mathematical modeling because they relate something’s rate of change to itself. For example, the most popular model for long-term population-change is a 

Expressed literally, Eq. 1 says that any function satisfying the conditions that the function is proportional to its first derivative and that 

 is proportional to its first derivative. This specific model will be discussed in more detail in section 6.

Differential equations are also very handy in finance. Specifically, differential equations are used examining stock investments. The value of a stock will change over time relative to how many people are willing to invest in the stock, which is also a function of the price of the stock. The changing relationship of a stock’s rate of change to itself screams “differential equation,” as the stock’s value changes over time in relation to the original price of the stock.

*This is true because the differential equation that describes the probability density of a particle’s position under a delta function potential gives non-zero probability solutions for a particle tunneling through an infinite-potential barrier (if confined to a single axis):

**We logically expect the position of a swinging pendulum to look like a cosine graph that decreases in height with time. This matches the solution under a certain set of parameters:

***The diffusion of diseases in humans is complicated because medical research and other social factors are variables that are computationally difficult to account for. However, for more vulnerable species, such as bananas, one would expect the area of effect of the disease to exponentially increase with time, which approximately matches statistical data.

****This differential equation is often more useful than the exponential differential equation because it accounts for population capacity. Notice that for 

, the differential equation has exponential solutions.

AP Calculus AB/BC Exams

AP Calculus AB/BC Exam Prep | Reviews +  Free Practice | Fiveable

AP Calculus AB/BC

Free Review 2020

Free Response Questions (FRQ)

Unit 1: Limits & Continuity

Unit 2: Differentiation: Definition & Fundamental Properties

Unit 3: Differentiation: Composite, Implicit & Inverse Functions

Unit 4: Contextual Applications of the Differentiation

Unit 5: Analytical Applications of Differentiation

Unit 6: Integration and Accumulation of Change

Unit 7: Differential Equations

Unit 8: Applications of Integration

Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)

Unit 10: Infinite Sequences and Series (BC Only)

Multiple Choice Questions (MCQ)

Blogs

Live Cram Sessions 2020

Unit 7 Overview: Differential Equations

Separable Differential Equations

Separable differential equations -- how to work with them, difference between general and particular solutions, some discussion of the domain of a solution of a differential equation, look at how separation of variables is used in the context of AP free response questions.

Separable Differential Equations Slides

testing possible solutions for differential equations

. The process is as simple as taking a derivative, plugging it into a formula, and then doing some simplifying to show that the solution does or does not work.

For example, we will test the following solution for the succeeding differential equation:

Following this step, we take the derivative by the 

 once, but we’ll use a technique that significantly cleans up the problem instead of jumping straight to differentiating again:

made in step 2 in Eq. 17 (as denoted by the 2 over the equality sign) is made possible by Eq. 16.

product rule to find the second derivative

The substitution made in step 3 in Eq. 18 is a combination of the 

product rule and implicit differentiation

The substitution made in step 1 is made possible by Eq. 17. This method may seem convoluted and unnecessary, but it seriously simplified some very tedious algebra.

From here, we can already see that the differential equation will

 it will hold for by plugging it into the differential equation put forth in Eq. 16:

The substitution made in step 2 is made possible by Eq. 16. The above condition holds only for 

Test if the function given in Eq. 20 is a solution to the differential equation in Eq. 21. Find an expression for the max value of 

. (Hint: use graphing software to find a shortcut to finding the max value of 

Verifying Solutions for Differential Equations

visualize a solution to a differential equation

 without actually solving the differential equation. Let’s construct a slope field to solidify this idea.

Slope fields essentially draw the slopes of line segments that go through certain points. Let’s consider the following differential equation:

, which we can put into a table for various coordinates:


We can use this data to draw an approximate solution to the differential equation by 

through each point that have the corresponding slope: 🌄

Sketching Slope Fields

The actual solution (which can actually be manipulated to be separable*) to the differential equation in Eq. 39 is the following:

https://www.desmos.com/calculator/fjli4efhcj

 and clicking the play button on equation 18, you can see that

 this curve does indeed fit the given slope field for any constant 

. Clicking the play button will show different curves for different values of 

The most intuitive way to think of a slope field is to picture a fluid flowing and then placing an object on the fluid that will trace out a path. This path is approximate to the solution to the curve that represents the differential equation. 

Fill in the table below for different values of 

 at different coordinate points. Use a calculator to find the values to two decimal places. Create a slope field and then solve the differential equation and confirm that your slope field matches the solution to the differential equation. ✍

Varying values of C plotted over the slope field are shown here:

https://www.desmos.com/calculator/ab0swk21ao

From here, one can solve the latter differential equation (which will give a solution that is only a function of 

, which will give the aforementioned solution to the original differential equation.

Reasoning Using Slope Fields

find the numerical values of functions based on a given differential equation and an initial condition

. We can approximate a function as a set of line segments using Euler’s method.

Before introducing this idea, it is necessary to understand two basic ideas.

This information allows us to do an algorithmic process to approximate function values when given a differential equation and an initial condition.

To showcase this method, let’s consider the following differential equation with a consequent initial condition:

. We will create a table that essentially creates that line-segment link in 

Notice that we can fill in the rest of the table and continue the process to get closer and closer to 

 is a constant value (which is typically called the step-size).

We can then fill in the rest of the table:

Note that as the step size approaches zero, the approximation becomes more and more exact. As an exercise, find an approximate value for 

Using Euler’s method, approximate the value of 

Then find the absolute error in the approximation by directly solving for 

 and compare the absolute error in this approximation to the original absolute error.

Approximating Solutions Using Euler’s Method

The separation of variables is a technique for finding 

 solutions to a differential equation. In this context, “general solutions” means that we will be solving for a 

 of functions. This may seem like abstract wording, but this idea has already been explored in integration and antidifferentiation: 💯

 is confined to a certain set of functions 

.” Or, more simply put, this means that there are infinite solutions to Eq. 22, but the solution must include the condition that 

 is a general solution. A specific solution for this same example would be: 

 is a specific number. Differential equations function in a similar way. First, find a general solution, then find a specific solution if you are given more information. 

7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

Finding General Solutions Using Separation of Variables

7.6 Finding General Solutions Using Separation of Variables

Let’s look at the differential equation given in Eq. 12:

, we can do something that would disgust professional mathematicians*:

We can perform step 5 because the two sides of the equation are equivalent, thus we can perform the same operation (ie. integration) on both sides and not lose any information.

Note that we need two constants because we found two different antiderivatives. However, we can 

This statement means that, because the set of real numbers is an infinite set** and both of the constants are real numbers, the difference between them is also a real number, ie. another constant we can call 

Notice that this equation is also equivalent to the solution given in Eq. 12. The solution given in Eq. 27 is a generalization of the solution given in Eq. 12.

Now, let’s say that we have some prior information about the function called an 

. We can use this information to solve for 

, which means we have found the specific solution to the differential equation in Eq. 12 along with the condition that 

We can also revisit the differential equation in Eq. 11:

We need to take special precautions regarding the 

. For an additional exercise, find a specific solution to the differential equation such that 

Another differential equation that is solvable via 

, the solution to the differential equation is just 

 into the differential equation in Eq. 33, you will notice that this returns a valid equality.

Solve the differential equation below given an initial condition and a range restriction. (Hint: rewrite the arcsine expression using the identity that defines a cosine graph as a shifted sine graph and do something similar for the arccosine expression.)

The arc trig expressions are red herrings:

*This mathematical move is actually incorrect because multiplying the 

 is a fraction, which is not true. However, this move is completely acceptable for the AP exam.

**“Infinite” in this context is not well-defined. The actual size of the set is called its cardinality, which is not simply infinity as implied in the statement:

Some definitions also include zero in the set of natural numbers (the lastly defined set in the statement).

Finding Particular Solutions Using Initial Conditions and Separation of Variables

 describe phenomena using a logistic differential equation:

term is often grouped together into a single constant.

The most popular (no pun intended) model using a logistic differential equation is modeling 

exponential growth as well as a population capacity

There are some properties we are going to derive that are necessary for the AP exam, in particular, the multiple-choice section.

The first one is the solution to the differential equation. It is separable but in a strange way. We will use the form of the differential equation as presented in Eq. 46:

This means we have to do a tough integral. As it is, we cannot integrate the left-hand side with normal techniques. We will assume that the function can be written in the following form:

This means the equation must be true for any value of 

 falls within the domain of the function.

The domain is of the function on the left-hand side in Eq. 47 is the following:

 to set up a system of equations to solve for 

Substituting Eq. 54 back into Eq. 48 yields a much easier integration:

. The best way to visualize this is by graphing the derived solution and seeing what happens as 

 gets infinitely large. In this case the graph approaches a 

The second value is a bit more difficult to solve. Let’s use implicit differentiation to find the second derivative of 

 using the form of the equation in Eq. 45:

Of course, this isn’t the full story, as this just means 

 is a critical point. You can verify this by finding the value of 

*This is not a maximum in the strict definition of the word; the function does not have a maximum but rather grows monotonically.

Unit 7: Differential Equations

AP Calculus AB/BC

all

videos

study guides

slides

hyper typer

browse by units

Free Review 2020

Free Response Questions (FRQ)

Unit 1: Limits & Continuity

Unit 2: Differentiation: Definition & Fundamental Properties

Unit 3: Differentiation: Composite, Implicit & Inverse Functions

Unit 4: Contextual Applications of the Differentiation

Unit 5: Analytical Applications of Differentiation

Unit 6: Integration and Accumulation of Change

Unit 7: Differential Equations

Unit 8: Applications of Integration

Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)

Unit 10: Infinite Sequences and Series (BC Only)

Multiple Choice Questions (MCQ)

Blogs

Unit 7: Differential Equations

AP Calculus AB/BC

all

videos

study guides

slides

hyper typer

browse by units

Free Review 2020﻿

Free Response Questions (FRQ)﻿

Unit 1: Limits & Continuity﻿

Unit 2: Differentiation: Definition & Fundamental Properties﻿

Unit 3: Differentiation: Composite, Implicit & Inverse Functions﻿

Unit 4: Contextual Applications of the Differentiation﻿

Unit 5: Analytical Applications of Differentiation﻿

Unit 6: Integration and Accumulation of Change﻿

Unit 7: Differential Equations﻿

Unit 8: Applications of Integration﻿

Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)﻿

Unit 10: Infinite Sequences and Series (BC Only)﻿

Multiple Choice Questions (MCQ)﻿

Blogs﻿

Free Review 2020

Free Response Questions (FRQ)

Unit 1: Limits & Continuity

Unit 2: Differentiation: Definition & Fundamental Properties

Unit 3: Differentiation: Composite, Implicit & Inverse Functions

Unit 4: Contextual Applications of the Differentiation

Unit 5: Analytical Applications of Differentiation

Unit 6: Integration and Accumulation of Change

Unit 7: Differential Equations

Unit 8: Applications of Integration

Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)

Unit 10: Infinite Sequences and Series (BC Only)

Multiple Choice Questions (MCQ)

Blogs