Sec 1.3 Differential Equations as Mathematical

ModelsSec 3.1 Linear Model

Where it appear

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Sec 1.3 Differential Equations as Mathematical ModelsSec 3.1 Linear Model

Behavior of some real-life system

or phenomenonMathematical Model

STEP 1: Identification of the variables

STEP 2: We make some reasonable assumptions (physical laws)

Construction of a Mathematical model:

AssumptionsMathematicalFormulation

ObtainSolutions

Check model Predictions with

Known facts

Express DE

Solve D

E

Display graphically

If necessary alter assum

ptions

1.Population Dynamics*2.Radioactive Decay*3.Spread of Disease*4.Chemical Reactions 5.Newton’s Law of Cooling*6.Mixtures*7.Draining a Tank8.Series Circuits9.Falling Bodies10.Falling Bodies and Air Resistance11.Suspended Cables

Applications

When a pizza is removed from an oven. Its temperature is measured at 300 F. Three minutes later its temperature is 200 F. How long will it take for the pizza to cool off to a room temperature of 70 F?

EXAMPLE:

Newton’s Law of Cooling-Warming: the rate at which the tempof a body changes is proportional to the difference between the temp of the body and the temp of the surrounding.

changes pizza theof tempheat which t the

tat time pizza theof temp time

dtdT

T(t)t

)( mTTdtdT

When a pizza is removed from an oven. Its temperature is measured at 300 F. Three minutes later its temperature is 200 F. How long will it take for the pizza to cool off to a room temperature of 70 F?

EXAMPLE:

Newton’s Law of Cooling-Warming: the rate at which the tempof a body changes is proportional to the difference between the temp of the body and the temp of the surrounding.

0 5 10 15 20 25 30 35 40

80

100

120

140

160

180

200

220

240

time in minutes

230 exp(-0.19018 t)+70

A thermometer is taken from a inside room to the outside, where the air temperature is 5F. After I minute the thermometer reads 55F, and after 5 minutes it reads 30F. What is the initial temperature of the inside room?

#14/p99:

Newton’s Law of Cooling-Warming: the rate at which the temp of a body changes is proportional to the difference between the temp of the body and the temp of the surrounding.

The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P_0 has doubled in 5 years, how long will it take to triple? Quadruple?

#1/p99:

Population: The population of a community is known to increase at a rate proportional to the number of people present at time t

Population: The population of a community is known to increase at a rate proportional to the number of people present at time t

2002023,500,000

2003124,250,000 0314.023502425ln

000,500,23

k

c

tetP 0314.0000,500,23)(

000,495,27)5(52007 Pt

Interested:See #39/101Read sec 3.2

Output rate

3 gal/min

input rate

3 gal/min

A large tank held 300 gallons of brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min. it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the 300 gallons, how much salt is in the tank after 10 min?

300 gallons

changes A(t)at which rate thedtdA

enterssalt at which rateinput inR

leavessalt at which rateoutput outR

tat timesalt ofamount )( tA

salt ofrateoutput

salt of

rateinput

dtdA

outin R R

Output rate

3 gal/min

input rate

3 gal/min

A large tank held 300 gallons of brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min. it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the 300 gallons, how much salt is in the tank after 10 min?

300 gallons

outin R R dtdA

brine ofrateinput

inflow

salt ofionconcentrat

R in

lb/min 6 gal/min 3 lb/gal 2 R in

brine ofrateoutput

outflow

salt ofionconcentrat

R out

gal/min 3 lb/gal R in ?

Output rate

3 gal/min

input rate

3 gal/min

A large tank held 300 gallons of brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min. it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow was 2 lb/gal. If 50 pounds of salt were dissolved initially in the 300 gallons, how much salt is in the tank after 10 min? how much salt in the tank after a long time?

300 gallons

100A 6

dtdA

100600t

CeA(t)

50)0( A

Sol:

100550600t

eA(t)

55050)0( CA

102.3394

10010

55060010

e)A(

0 100 200 300 400 500 600

100

150

200

250

300

350

400

450

500

550

600

Time (minutes)

Amountof

Salt A

Output rate

4 L/min

input rate

4 L/min

A tank held 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of slat per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.

200 liters

#19p99

Chapter-Summary Chapter # 1 and Chapter # 2

Next week (mond –wed)

GROUPMembersChapter Date of submission

Expected Datefor presentation

GROUP 1

1)AL-GHAMDI, MOHAMMAD TURKI2)AL-ZAHRANI, AHMAD ALI AHMAD3)L-HADI, MOHAMMAD SALEH HADAJ4)AL-MUTAIRI, ABDUL-LATIF THUWAINI

1Mon 2/25/2008  11:58 AM 

GROUP 2

1)Khalid Abdulghani2)Khaldoon Al-Azzah3)Yousef Al-Shaheen4)Abdulrahman Al-Saggaf5)Mohammad Bawazeer6)Ibraheem Alsufyani

2Mon 2/25/2008  12:24 PM