Unit 7: Differential Equations

Learning Target 7-1: Modeling Situations with Differential Equations

Objective: Interpret verbal statements of problems as differential equations

involving a derivative expression.

The rate at which the temperature of coffee changes can be modeled by ��

��� 0.05470 � � , where C

represents the temperature of the coffee, in ˚Fahrenheit, and t is measured in minutes since the coffee was

poured.

1. How fast is the temperature of the coffee changing when the coffee is first poured, at a temperature of

185˚? How fast is the temperature of the coffee changing when the coffee is 170˚?

2. Explain why the graph of �� cannot resemble either of the following two graphs.

3. We will soon learn how to show that �� � 70 � 115���.����.

a. What is �0 ? What does this number represent?

b. Find lim�→� �� . Interpret this limit in the context of this problem.

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4. Find �′� .

5. Show that your answer in question 4 is equivalent to ��

�� given above.

6. Ms. Cherestal pours her cup of coffee but then remembers that she still has to go and make copies. If

she wants the coffee to stay hot as long as possible, should she pour in the cold milk and then go and

make copies or should she wait until she comes back to pour in the cold milk? Explain your reasoning.

Important Ideas:

Check Your Understanding!

1. The number of fruit flies increases at a rate proportional to its current population, F. Write a

differential equation to represent this situation.

2. The weight of a baby bird at birth (� � 0 is 20 grams. If �� is the weight of the bird, in grams, t days

after it is first weighed, then ��

���

�

�100 � � .

a. At what rate is the bird’s weight changing when it weighs 30 g? When it weighs 90 g?

b. What can you say about the rate of change of the bird’s weight as it gets closer to its adult

weight (100 g)?

c. Explain why the graph shown can NOT be the graph of �� .

3. Which of the following functions are solutions to the differential equation � � � � 0?

(Choose all that apply.)

A) � � �!

B) � � ��!

C) � � sin $

D) � � �cos $

E) � � sin3$

F) � � 3 cos $

myAP Examples:

Ex. 1: For what value of (, if any, will � � ( sin5$ � 2 cos4$ be a solution to the differential equation

�" � 16� � �27sin5$ ?

Ex. 2: If the pressure , applied to a gas is increased while the gas is held at a constant temperature, then the

volume - of the gas will decrease. The rate of change of the volume of gas with respect to the pressure is

proportional to the reciprocal of the square of the pressure. Which of the following is a differential equation

that could describe this relationship?

Learning Target 7-2: Verifying Solutions for Differential Equations.

Objective: Verify solutions to differential equations.

***The material for this lesson is incorporated in 7-1 Notes***

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Learning Target 7-3: Sketching Slope Fields.

Objective: Estimate solutions to differential equations.

A slope field is a pictorial representation of all of the possible solutions to a given differential equation.

Remember that a differential equation is the first derivative of a function, )(' xf or dx

dy. Thus, the solution

to a differential equation is the function, f(x) or y.

There is an infinite number of solutions to a differential equation. Why?

For the AP Exam, you are expected to be able to do the following four things with slope fields:

1.________________________________________________________________________________

________________________________________________________________________________

2.________________________________________________________________________________

________________________________________________________________________________

3.________________________________________________________________________________

________________________________________________________________________________

4.________________________________________________________________________________

________________________________________________________________________________

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#1 Sketch a slope field for a given differential equation.

Given the differential equation below,

compute the slope for each point indicated on

the grid to the right.

Then, make a small mark that approximates

the slope through the point.

Given the differential equation below,

compute the slope for each point

Indicated on the grid to the right.

Then, make a small mark that approximates

the slope through the point.

When you are done, add a point between each

of the current ones. Calculate the slope for

every additional point and draw a small

marking that approximates the slope.

1dy

xdx

= −

dyx y

dx= −

#2 Given a slope field, sketch a solution curve through a given point.

To the right is pictured the slope field that you

developed for the differential equation

on the previous page.

Sketch the solution curve through the point (1, -

1).

To do this, you find the point and then follow

the slopes to approximate what the graph could

look like.

1dy

xdx

= −

#3 Match a slope field to a differential equation.

Since the slope field represents all of the particular solutions to a differential

equation, and the solution represents the ANTIDERIVATIVE of a differential

equation, then the slope field should take the shape of the antiderivative of dy/dx.

Match the slope fields to the differential equations on the next page.

A. B. C.

D. E. F.

G. H. I.

J.

1. xdx

dysin=

2. 42 += xdx

dy

3. xe

dx

dy=

4. 2=dx

dy

5. xxdx

dy33 −=

6. xdx

dycos2=

7. xdx

dy24 −=

8. xdx

dy=

9. dy

dx= x2

10. xdx

dy 1−=

#4 More Practice: Matching a slope field to a solution to a differential equation.

When given a slope field and a solution to a differential equation, then the slope

field should look like the solution, or y.

Match the slope fields below to the solutions on the next page.

A. B. C.

D. E. F.

G. H. I.

J.

1. xy =

2. 2xy =

3. xey =

4. 2

1

xy =

5. 3xy =

6. xy sin=

7. xy cos=

8. xy =

9. 1=y

10. xy tan=

Shown below is a slope field for which of the following differential equations? Explain your reasoning for each

of the choices below.

Consider the differential equation y

x

dx

dy= to answer the following questions.

a. On the axes below, sketch a slope field for the equation.

b. Sketch a solution curve that passes through the point (0, –1) on your slope field.

c. Find the particular solution y = f(x) to the differential equation with the initial condition f(0) = –1.

myAP Questions:

Ex. 1:

The slope field for a certain differential equation is shown above. Which of the following statements about a

solution � � .$ to the differential equation must be false.

A. The graph of the particular solution that satisfies .�3 � 2 is concave up on the interval �3 < $ < 3.

B. The graph of the particular solution with .�2 � �2 has a relative minimum at $ � �2.

C. The graph of the particular solution that satisfies .0 � �2 is concave up on the interval �1 < $ < 3.

D. The graph of the particular solution that satisfies .1 � 0 is linear.

Ex. 2:

The slope field for a certain differential equation is shown above. Which of the following could be a solution

to the differential equation with initial condition 0 � 0 ?

A. � � ln|1 � $|

B. � � ��!1� 1

C. � �!

��!

D. � �!1

��!1

Learning Target 7-4: Reasoning Using Slope Fields.

Objective: Estimate solutions to slope fields.

***The material for this lesson is incorporated in 7-3 Notes***

7-4

Learning Target 7-5: Approximating Solutions using Euler’s Method.

Objective:

***This lesson is for AP Calc BC only***

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Learning Target 7-6: Finding General Solutions Using Separation of Variables.

Objective: Determine general solutions to differential equations.

1. Let �2

�!� $�. The slope field is shown below.

a. Describe the slopes of the tangent lines in

each quadrant.

b. Using three different colored pencils, sketch three

possible solution curves to this differential equation.

c. Describe the general shape of a solution curve.

d. How do the solution curves differ? Why do you think this is?

2. Let’s think about how to find the solution analytically by looking at an easier example.

a. What is the general solution to �2

�!� $? How do you know?

b. Explain why 4� � $4$ is equivalent to �2

�!� $.

c. What is 5 4� and what is 5$4$?

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3. Let’s go back to our original problem.

a. Re-write �2

�!� $� so all the terms with y are on the left side and all the terms with x are on the

right side.

b. Take the integral of both sides.

c. Get � by itself. How does the algebra support what you found in 1d?

Check Your Understanding!

4. Use separation of variables to find the general solution to the differential equation.

a. �2

�!� 3� b.

�2

�!�

!

2

5. What is the general solution to the differential equation �2

�!�

678!:;<=>

6782?

6. What is the general solution to the differential equation �2

�!� �? � �? sin $?

7.

8.

Learning Target 7-7: Finding Particular Solutions Using Initial Conditions.

Objective: Determine particular solutions to differential equations.

Given below are differential equations with given initial condition values. Find the particular solution for

each differential equation.

1. 266 2 ++= xxdx

dy and f(–1) = 2

2. x

x

dx

dy

2

121 23

+= and f(0) = 2

3. y

xx

dx

dy

2

22 += and f(0) = 2 4.

y

x

dx

dy 2+= and f(1) = –3

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5. )2(4 −= yxdx

dyand f(0) = 0

6. 2

1

x

y

dx

dy −= and f(2) = 0

Non Calculator

The velocity of a particle is given by the function v(t) = 3t + 6 and the position at t = 2 is 3.

a. Is the particle moving to the left or right at t = 2? Justify your answer.

b. What is the position of the particle, s(t), for any time t > 0?

c. Does the particle ever change directions? Justify your answer.

d. Find the total distance traveled by the particle for t = 1 to t = 4.

Non Calculator

Non Calculator

The acceleration of a particle moving along the x – axis at time t is given by a(t) = 6t – 2. If the velocity is 25

when t = 3 and the position is 10 when t = 1, then the position x(t) =

A. 9t2 + 1

B. 3t2 – 2t + 4

C. t3 – t

2 + 4t + 6

D. t3 – t

2 + 9t – 20

E. 36t3 – 4t

2 – 77t + 55

A particle moves along the x-axis so that, at any time t > 0, its acceleration is given by a(t) = 6t + 6. At time t = 0, the

velocity of the particle is –9 and its position is –27.

a. Find v(t), the velocity of the particle at any time t.

b. For what value(s) of t > 0 is the particle moving to the right? Justify your answer.

c. Find the net distance traveled by the particle over the interval [0, 2].

d. Find the total distance traveled by the particle over the interval [0, 2].

Calculator

A particle moves along the x – axis so that its velocity is given by the function5

1cos2

)( −=t

ttv . On the interval

0 < t < 10, how many times does the particle change directions?

A. One

B. Three

C. Four

D. Five

E. Seven

Learning Target 7-8: Exponential Models with Differential Equations.

Objective: Interpret the meaning of a differential equation and its variables in

context.

Exponential Growth and Decay Models: In many applications, the rate of change of a variable, y, is

proportional to the value of y. If y is a function of time, t, the proportion can be written as: �2

��� (�. The general

solution to this type of differential equation is: � � @�A� where A is the initial value of y, and k is the

proportionality constant.

Exponential GROWTH occurs when k > 0 Exponential DECAY occurs when k < 0

1. The rate of change of y is proportional to y. When t = 0, y = 2. When t = 2, y = 4. What is the value of y

when t = 3?

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2. 1988 AP Calculus BC #43

Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria

doubles in three hours, in how many hours will the number of bacteria triple?

A) 3ln3

ln2 B)

2 ln3

ln2 C)

ln3

ln2 D) ln

27

2

E) ln

9

2

3. 1993 AP Calculus BC #38

During a certain epidemic, the number of people that are infected at any time increases at a rate proportional

to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first

discovered, and 1,200 are infected 7 days later, how many people are infected 12 days after the epidemic is

first discovered?

A) 343 B) 1,343 C) 1,367 D) 1,400 E) 2,057

4. 1993 AP Calculus AB #42

A puppy weights 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its

first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it

is 3 months old?

A) 4.2 pounds B) 4.6 pounds C) 4.8 pounds D) 5.6 pounds E) 6.5 pounds

5. 2011 AP Free Response Question #5 -

Solve each of the following separable differential equations:

6. �2

�!� 5$� cos � cot � and�0 � 0 7.

�2

�!� �2$ sin? � and�0 � 4