Mathematical Modeling with Differential Equations

Chapter 9: By, Will Alisberg Edited By Emily Moon

Overview

  9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method  9.3 Modeling with First-Order Differential Equations  Quiz

Overview

  9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method  9.3 Modeling with First-Order Differential Equations  Quiz

Key Definitions

 Differential Equation- Any equation in which the derivative affects the f(x)… e.g. f(x)=f’(x)/(2x)  Order- the highest degree of differentiation in a differential equation  Integral Curve- Graph of a solution of a differential equation

First Order Initial Value Problems

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

 Replace variables to solve



General Solution

  Start by Converting to: Calculate  x)  Use General Solution:

dy dx



p

(

x

)

y



q

(

x

)  (

x

) 

e P

(

x

)

y

   ( 1

x

)  

q

(

x

)

My Turn!

dy dx



x

3 

y

 4

y dy

 5

y



dx p

(

x

)   5

x

3

q

(

x

) 

x

3

P

(

x

)   5

x

 (

x

) 

e

 5

x y

 So… 1

e

 5

x



e

 5

x

(

x

3 )

dx

Set up the integral for the given differential equation

Your Turn!

Set up the integral to solve for y

x

2

dy dx



dy dx



x

2

y



x

 1 

y

Wonhee Lee (

x

2  1 )(

dy dx



y

) 

x

 1

dy



y



dx p

(

x

)  1

x

1  1

q

(

x

) 

P

(

x

) 

x

1  1

x

 (

x

) 

e x y

 1

e x



e x x

 1

Newton’s Second Law

Overview

  9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method  9.3 Modeling with First-Order Differential Equations  Quiz

Key Definitions

     Direction Field- A graph showing the slope of a function at each point Euler’s Method- A technique for obtaining approximations of f(x) Absolute Error- Difference between approximated value of f(x) and actual value Percentage error- Absolute Error divided by the Exact value of f(x), Multiply the decimal by 100 to obtain a percentage Iteration- One cycle of a method such as Newton’s or Euler’s

Direction Field

 Show Slopes at Various Points on a Graph  Follow the trail of lines   Different arrows with the same value of x represent different c’s Don’t forget the points on the axes

Euler’s Method: Theory

 Approximates values of f(x) through small changes in x and its derivative   The algebraic idea behind slope fields More 

x

make a more accurate approximation

Euler’s Method: Calculation

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

Use

y

1 

y

0 

f

 (

x

0 )( 

x

)

x

1 

x

0  

x

Repeat  Note: with very small values of 

x

we will get

y



y

0 

f

 (

x

)

dx

Your Turn!

With a step size of  1 approximate Knowing

dy

 3

dx x

:

x

 4 Wonhee Lee

y

( 1 )  4 Just kidding- Go ahead Anna

y

 4  3  1 .

5  1  .

75  10 .

25

Overview

  9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method  9.3 Modeling with First-Order Differential Equations  Quiz

Key Defintions

     Uninhibited growth model- y(x) will not have a point at which it will not be defined Carrying Capacity- The magnitude of a population an environment can support Exponential growth- No matter how large y is, it will grow by a% in the same amount of time Exponential decay- No matter how large y is, it will decrease by b% in the same amount of time Half-Life- The time it takes a population to reduce itself to half its original size

Exponential Growth and Decay

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

y



y

0

e kt

My Turn!

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

y

 64000

e kt

3 

e

6

k k

 ln 3 6

y



64000

e

2 ln 3

y



576000

Your Turn!

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

Jiwoo, from Maryland

Answer

y



y

0

e kt

.

5 

e

2 .

2

k k

 ln .

5 2 .

2

t

.

1  2

e t

(ln .

5 ) 2 .

2  9 .

508

Overview

  9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method  9.3 Modeling with First-Order Differential Equations  Quiz

Quiz!

  1.

If a substance decomposes at a rate proportional to the substance present, and the amount decreases from 40 g to 10 g in 2 hrs, then the constant of proportionality (k) is A. -ln2 B. -.5 C -.25 D. ln (.25) E. ln (.125) 2. The solution curve of

y

 (

x

) 

y

that A. D.

y y

passes through the point (2,3) is  

e x e x

  (

e

2 3  B. 3) E.

y



y

2

x

 

e x

5 .406

C.

y

 .406

e x

   

More Quiz Questions

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

 A stone is thrown at a target so that its velocity after t seconds is (100-20t) ft/sec. If the stone hits the target in 1 sec, then the distance from the sling to the target is: A. 80 ft B. 90 ft C. 100 ft D. 110 ft E. 120 ft

Last Quiz Question

 If you use Euler’s method with = .1 for the differential equation y’(x)=x with the initial value y(1)=5, then, when x= 1.2, y is approximately:  A. 5.10 B. 5.20 C. 5.21 D. 6.05 E. 7.10

 1A  2C  3True  4B  5C

Quiz Answers

Bibliography

    Barron’s “How to Prepare for the Advanced Placement Exam: Calculus Anton, Bivens, Davis “Calculus” http://exploration.grc.nasa.gov/education/rocket/Images/newto n2r.gif

http://www.usna.edu/Users/math/meh/euler.html