Mathematical Modeling with Mathematical Modeling with Differential EquationsDifferential Equations

Chapter 9: By, Will AlisbergChapter 9: By, Will Alisberg

Edited By Emily MoonEdited By Emily Moon

OverviewOverview

9.1 First-Order Differential Equations and 9.1 First-Order Differential Equations and ApplicationsApplications

9.2 Direction Fields; Euler’s Method9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential 9.3 Modeling with First-Order Differential

EquationsEquations QuizQuiz

OverviewOverview

9.1 First-Order Differential Equations and 9.1 First-Order Differential Equations and ApplicationsApplications

9.2 Direction Fields; Euler’s Method9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential 9.3 Modeling with First-Order Differential

EquationsEquations QuizQuiz

Key DefinitionsKey Definitions

Differential Equation- Any equation in which Differential Equation- Any equation in which the derivative affects the f(x)… e.g. the derivative affects the f(x)… e.g. f(x)=f’(x)/(2x)f(x)=f’(x)/(2x)

Order- the highest degree of differentiation in Order- the highest degree of differentiation in a differential equationa differential equation

Integral Curve- Graph of a solution of a Integral Curve- Graph of a solution of a differential equationdifferential equation

First Order Initial Value ProblemsFirst Order Initial Value Problems

Find a general formula Find a general formula for y(x) and use initial for y(x) and use initial condition to solve for C.condition to solve for C.

Replace variables to Replace variables to solvesolve

General SolutionGeneral Solution

Start by Converting to:Start by Converting to: Calculate Calculate x)x) Use General Solution:Use General Solution:

)()( xqyxpdx

dy

y 1

(x)q(x)

(x) eP (x )

My Turn!My Turn!

xex

xxP

xxq

xp

xydx

dy

yyxdx

dy

5

3

3

3

)(

5)(

)(

5)(

5

4

So…

dxxee

y xx

)(1 35

5

Set up the integral for the given differential equation

Your Turn!Your Turn!

1

1

)(

)(1

1)(

1)(1

1

1))(1( 2

x

e

ey

ex

xxPx

xq

xpx

ydx

dy

xydx

dyx

x

x

x

Set up the integral to solve for y

Wonhee Lee

yxyxdx

dy

dx

dyx 122

Newton’s Second LawNewton’s Second Law

OverviewOverview

9.1 First-Order Differential Equations and 9.1 First-Order Differential Equations and ApplicationsApplications

9.2 Direction Fields; Euler’s Method9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential 9.3 Modeling with First-Order Differential

EquationsEquations QuizQuiz

Key DefinitionsKey Definitions Direction Field- A graph showing the slope of a Direction Field- A graph showing the slope of a

function at each pointfunction at each point Euler’s Method- A technique for obtaining Euler’s Method- A technique for obtaining

approximations of f(x)approximations of f(x) Absolute Error- Difference between approximated Absolute Error- Difference between approximated

value of f(x) and actual valuevalue of f(x) and actual value Percentage error- Absolute Error divided by the Exact Percentage error- Absolute Error divided by the Exact

value of f(x), Multiply the decimal by 100 to obtain a value of f(x), Multiply the decimal by 100 to obtain a percentagepercentage

Iteration- One cycle of a method such as Newton’s or Iteration- One cycle of a method such as Newton’s or Euler’sEuler’s

Direction FieldDirection Field

Show Slopes at Various Show Slopes at Various Points on a GraphPoints on a Graph

Follow the trail of linesFollow the trail of lines Different arrows with the Different arrows with the

same value of x represent same value of x represent different c’sdifferent c’s

Don’t forget the points Don’t forget the points on the axeson the axes

Euler’s Method: TheoryEuler’s Method: Theory

Approximates values of Approximates values of f(x) through small f(x) through small changes in x and its changes in x and its derivativederivative

The algebraic idea The algebraic idea behind slope fieldsbehind slope fields

More More make a more make a more accurate approximationaccurate approximation

x

Euler’s Method: CalculationEuler’s Method: Calculation

Starting with a known point on Starting with a known point on a function, knowing the a function, knowing the equation for the function.equation for the function.

Use Use

RepeatRepeat Note: with very small values of Note: with very small values of

we will get we will get

xxx

xxfyy

01

001 ))((

x dxxfyy )(0

Your Turn!Your Turn!

25.1075.15.134 y

4)1( y

With a step size of With a step size of approximate approximate

Knowing Knowing 4:3

xxdx

dy

1x

Wonhee Lee

Just kidding- Go ahead Anna

OverviewOverview

9.1 First-Order Differential Equations and 9.1 First-Order Differential Equations and ApplicationsApplications

9.2 Direction Fields; Euler’s Method9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential 9.3 Modeling with First-Order Differential

EquationsEquations QuizQuiz

Key DefintionsKey Defintions

Uninhibited growth model- y(x) will not have a point Uninhibited growth model- y(x) will not have a point at which it will not be definedat which it will not be defined

Carrying Capacity- The magnitude of a population an Carrying Capacity- The magnitude of a population an environment can supportenvironment can support

Exponential growth- No matter how large y is, it will Exponential growth- No matter how large y is, it will grow by a% in the same amount of timegrow by a% in the same amount of time

Exponential decay- No matter how large y is, it will Exponential decay- No matter how large y is, it will decrease by b% in the same amount of timedecrease by b% in the same amount of time

Half-Life- The time it takes a population to reduce Half-Life- The time it takes a population to reduce itself to half its original sizeitself to half its original size

Exponential Growth and DecayExponential Growth and Decay

kteyy 0Where k is a constant, if k is negative, y will decrase, if k is positive, y will increase

My Turn!My Turn!

The bacteria in a certain The bacteria in a certain culture continuously culture continuously increases so that the increases so that the population triples every population triples every six hours, how many six hours, how many will there be 12 hours will there be 12 hours after the population after the population reaches 64000?reaches 64000?

6

3ln

3

640006

k

e

eyk

kt

576000

64000 3ln2

y

ey

Your Turn!Your Turn!

The concentration of Drug Z in a bloodstream The concentration of Drug Z in a bloodstream has a half life of 2 hours and 12 minutes. Drug has a half life of 2 hours and 12 minutes. Drug Z is effective when 10% or more of one tablet Z is effective when 10% or more of one tablet is in a bloodstream. How long after 2 tablets of is in a bloodstream. How long after 2 tablets of Drug Z are taken will the drug become Drug Z are taken will the drug become inaffective?inaffective?

Jiwoo, from Maryland

AnswerAnswer

508.9

21.

2.2

5.ln

5.

2.2

)5.(ln

2.2

0

t

e

k

e

eyy

t

k

kt

OverviewOverview

9.1 First-Order Differential Equations and 9.1 First-Order Differential Equations and ApplicationsApplications

9.2 Direction Fields; Euler’s Method9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential 9.3 Modeling with First-Order Differential

EquationsEquations QuizQuiz

Quiz!Quiz!

1.1. If a substance decomposes at a rate If a substance decomposes at a rate proportional to the substance present, and the proportional to the substance present, and the amount decreases from 40 g to 10 g in 2 hrs, amount decreases from 40 g to 10 g in 2 hrs, then the constant of proportionality (k) isthen the constant of proportionality (k) is

A. -ln2 B. -.5 C -.25 D. ln (.25) E. ln (.125)A. -ln2 B. -.5 C -.25 D. ln (.25) E. ln (.125)2. The solution curve of 2. The solution curve of that passes that passes

through the point (2,3) isthrough the point (2,3) isA. A. B. B. C. C.D. D. E. E.

y (x) y

y ex 3

y 2x 5

y .406ex

y ex (e2 3)

y ex

.406

More Quiz QuestionsMore Quiz Questions

True or False? If the second derivative of a True or False? If the second derivative of a function is a constant positive number, Euler’s function is a constant positive number, Euler’s Method will approximate a number smaller Method will approximate a number smaller than the true value of y?than the true value of y?

A stone is thrown at a target so that its velocity A stone is thrown at a target so that its velocity after t seconds is (100-20t) ft/sec. If the stone after t seconds is (100-20t) ft/sec. If the stone hits the target in 1 sec, then the distance from hits the target in 1 sec, then the distance from the sling to the target is:the sling to the target is:

A. 80 ft B. 90 ft C. 100 ft D. 110 ft E. 120 ftA. 80 ft B. 90 ft C. 100 ft D. 110 ft E. 120 ft

Last Quiz QuestionLast Quiz Question

If you use Euler’s method with If you use Euler’s method with = .1 for the = .1 for the differential equation y’(x)=x with the initial differential equation y’(x)=x with the initial value y(1)=5, then, when x= 1.2, y is value y(1)=5, then, when x= 1.2, y is approximately:approximately:

A. 5.10 B. 5.20 C. 5.21 D. 6.05 E. 7.10A. 5.10 B. 5.20 C. 5.21 D. 6.05 E. 7.10

x

Quiz AnswersQuiz Answers

1A 1A 2C 2C 3True 3True 4B4B 5C5C

BibliographyBibliography

Barron’s “How to Prepare for the Advanced Placement Exam: Barron’s “How to Prepare for the Advanced Placement Exam: CalculusCalculus

Anton, Bivens, Davis “Calculus”Anton, Bivens, Davis “Calculus” http://exploration.grc.nasa.gov/education/rocket/Images/newtohttp://exploration.grc.nasa.gov/education/rocket/Images/newto

n2r.gifn2r.gif http://www.usna.edu/Users/math/meh/euler.htmlhttp://www.usna.edu/Users/math/meh/euler.html