Sec 1.3 Differential Equations as Mathematical Models

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Sec 1.3 Differential Equations as Mathematical
Models
Sec 3.1 Linear Model
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Where it appear
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Mechanical Eng.
Where it appear
Dr. Faisal fairag
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Chemical Eng.
Where it appear
Dr. Faisal fairag
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Electrical Eng.
Where it appear
Dr. Faisal fairag
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Civil Eng.
Where it appear
Dr. Faisal fairag
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Heat Transfer
Where it appear
Dr. Faisal fairag
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Civil Eng.
Where it appear
Dr. Faisal fairag
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Where it appear
Dr. Faisal fairag
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Sec 1.3 Differential Equations as Mathematical
Models
Sec 3.1 Linear Model
Behavior of some real-life system or phenomenon
Mathematical Model
Construction of a Mathematical model
STEP 1 Identification of the variables
STEP 2 We make some reasonable assumptions
(physical laws)
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Mathematical Formulation
Assumptions
Express DE
Solve DE
If necessary alter assumptions
Check model Predictions with Known facts
Obtain Solutions
Display graphically
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Applications

1. Population Dynamics
2. Radioactive Decay
3. Spread of Disease
4. Chemical Reactions
5. Newtons Law of Cooling
6. Mixtures
7. Draining a Tank
8. Series Circuits
9. Falling Bodies
10. Falling Bodies and Air Resistance
11. Suspended Cables

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EXAMPLE
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?
Newtons 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.
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EXAMPLE
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?
Newtons 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.
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14/p99
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?
Newtons 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.
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1/p99
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?
Population The population of a community is
known to increase at a rate proportional to the
number of people present at time t
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Population The population of a community is
known to increase at a rate proportional to the
number of people present at time t
23,500,000 0 2002
24,250,000 1 2003
Interested See 39/101 Read sec 3.2
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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?
input rate 3 gal/min
300 gallons
Output rate 3 gal/min
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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?
input rate 3 gal/min
300 gallons
Output rate 3 gal/min
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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?
input rate 3 gal/min
Sol
300 gallons
Output rate 3 gal/min
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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.
19 p99
input rate 4 L/min
200 liters
Output rate 4 L/min
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Chapter-Summary
Chapter 1 and Chapter 2
Next week (mond wed)
Expected Date for presentation Date of submission Chapter Members GROUP
  Mon 2/25/2008  1158 AM 1 1)AL-GHAMDI, MOHAMMAD TURKI 2)AL-ZAHRANI, AHMAD ALI AHMAD 3)L-HADI, MOHAMMAD SALEH HADAJ 4)AL-MUTAIRI, ABDUL-LATIF THUWAINI GROUP 1
  Mon 2/25/2008  1224 PM 2 1)Khalid Abdulghani 2)Khaldoon Al-Azzah 3)Yousef Al-Shaheen 4)Abdulrahman Al-Saggaf 5)Mohammad Bawazeer 6)Ibraheem Alsufyani GROUP 2