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SPECIALIST MATHS

This week's specialist maths lesson focuses on problem solving with separable differential equations. Topics include finding external temperature, decay rates, terminal velocity, and growth & decay problems. Solutions and examples are provided for better understanding.

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SPECIALIST MATHS

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  1. SPECIALIST MATHS Differential Equations Week 3

  2. Problem Solving with Separable Differential Equations

  3. Example 1 (Ex 8E1) • Fluid is flowing through thick walled tube. • It maintains the temperature on the inner wall of the pipe at 600oC. • Heat is lost through the tube per unit length according to the formula: • Calculate the external temperature of the tube. r 0.02m 0.04m

  4. Solution to Example 1

  5. Solution to Example 1 continued

  6. Solution to Example 1 continued again

  7. Example 2 (Ex 8E2) • The rate of decay of a radioisotope is proportional to the number of particles present. • If the number of particles decreases by 10% after 50 years, find the percentage left after 200 years.

  8. Solution to example 2

  9. Solution to example 2 continued

  10. Solution to example 2 continued again

  11. Example 3 (Ex 8E2) • A man falls out of a plane and accelerates toward the earth with an acceleration given by • Terminal velocity is a constant velocity he will eventually reach. • Calculate his terminal velocity.

  12. Solution to Example 3

  13. Solution to Example 3 continued

  14. Example 4 (Ex 8E2) • The rate at which temperature changes is proportional to the differences between its temperatures. • A hot metal bar is 700oC at 12:00pm when taken out of the furnace. Its temperature falls to 200oC at 2:00pm. • If the air temperature is 20oC, calculate its temperature at 3:00pm.

  15. Solution to Example 4

  16. Solution to Example 4

  17. Solution to Example 4 continued

  18. Growth and Decay Problems

  19. Growth & Decay Graphs y y x x

  20. Logistic Differential Equations • The function starts off growing exponentially, then flattens out approaching a limiting value “A”. • This happens with populations as the recourses available to support it are limited.

  21. Logistic Differential Equations • Logistic differential equations are of the form: • When P is small, an exponential growth. • When a horizontal line.

  22. A Handy Algebraic Manipulation

  23. Example 5 (Ex 8F) The relative growth rate of snakes on Groute Island is given by: (a) What is the environments carrying capacity? (b) Solve the differential equation if there are 2280 snakes originally. (c) When would you expect the population to reach 4000 snakes.

  24. Solution to Example 5

  25. Example 6 (Ex 8F) Returning from a visit to a remote planet, Captain Boss brings with him a rare disease that infects the crew of the ship, spreading according to a logistic equation. (1) If there are 126 crew members write a differential equation governing the spread. (2) After 3 days 15 members have the disease, write a formula for the spread in terms of t. (3) When is the rate of infection greatest.

  26. (1)

  27. (2) When t = 0, N = 1

  28. When t = 3, N = 15

  29. (3) When is the rate of infection greatest.

  30. This Week • Text book pages 294 – 307. • Exercise 8E2 Q1 – 12 • Exercise 8F Q1 - 4. • Review Sets 8A -8D

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