1. Sec 1.3 Differential Equations as Mathematical Models Sec 3.1 Linear Model

2. Where it appear

3. Mechanical Eng. Dr. Faisal fairag Where it appear

4. Chemical Eng. Dr. Faisal fairag Where it appear

5. Electrical Eng. Dr. Faisal fairag Where it appear

6. Civil Eng. Dr. Faisal fairag Where it appear

7. Heat Transfer Dr. Faisal fairag Where it appear

8. Civil Eng. Dr. Faisal fairag Where it appear

9. Dr. Faisal fairag Where it appear

10. Sec 1.3 Differential Equations as Mathematical Models Sec 3.1 Linear Model Behavior of some real-life system or phenomenon Mathematical Model STEP 1: Identification of the variables STEP 2: We make some reasonable assumptions (physical laws) Construction of a Mathematical model:

11. Assumptions Mathematical Formulation Obtain Solutions Check model Predictions with Known facts Express DE Solve DE Display graphically If necessary alter assumptions

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

13. 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.

14. 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.

15. 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.

16. 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

17. Population: The population of a community is known to increase at a rate proportional to the number of people present at time t 23,500,00002002 24,250,00012003 Interested: See #39/101 Read sec 3.2

18. 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

19. 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 ?

20. 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 Sol:

21. 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 #19 p99

22. Chapter-Summary Chapter # 1 and Chapter # 2 Next week (mond –wed) Expected Date for presentation Date of submission ChapterMembersGROUP Mon 2/25/2008 11:58 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 12:24 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