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Sec 1.3 Differential Equations as Mathematical Models

Sec 1.3 Differential Equations as Mathematical Models. Sec 3.1 Linear Model. Where it appear. Mechanical Eng. Where it appear. Dr. Faisal fairag. Chemical Eng. Where it appear. Dr. Faisal fairag. Electrical Eng. Where it appear. Dr. Faisal fairag. Civil Eng. Where it appear.

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Sec 1.3 Differential Equations as Mathematical Models

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

  2. Where it appear

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

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

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

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

  7. Heat Transfer Where it appear Dr. Faisal fairag

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

  9. Where it appear Dr. Faisal fairag

  10. 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)

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

  12. Applications • Population Dynamics* • Radioactive Decay* • Spread of Disease* • Chemical Reactions • Newton’s Law of Cooling* • Mixtures* • Draining a Tank • Series Circuits • Falling Bodies • Falling Bodies and Air Resistance • Suspended Cables

  13. 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? 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. 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? 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. #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? 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. #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

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

  18. 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

  19. ? 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

  20. 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

  21. 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

  22. Chapter-Summary Next week (mond –wed) Chapter # 1 and Chapter # 2

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