Understanding Ordinary Differential Equations - Physical Examples and Methods

1. Ordinary Differential EquationsEverything is ordinary about them

2. Popping tags means • Popping bubble wrap • Using firecrackers • Changing tags of regular items in a store with tags from clearance items • Taking illicit drugs

3. Physical Examples

4. How long will it take to cool the trunnion?

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6. What did I learn in the ODE class?

7. In the differential equation the variable x is the variable • Independent • Dependent

8. In the differential equation the variable yis the variable • Independent • Dependent

9. Ordinary differential equations can have these many dependent variables. • one • two • any positive integer

10. Ordinary differential equations can have these many independent variables. • one • two • any positive integer

11. A differential equation is considered to be ordinary if it has • one dependent variable • more than one dependent variable • one independent variable • more than one independent variable

12. Classify the differential equation • linear • nonlinear • undeterminable to be linear or nonlinear

13. Classify the differential equation • linear • nonlinear • linear with fixed constants • undeterminable to be linear or nonlinear

14. Classify the differential equation • linear • nonlinear • linear with fixed constants • undeterminable to be linear or nonlinear

15. The velocity of a body is given by Then the distance covered by the body from t=0 to t=10 can be calculated by solving the differential equation for x(10) for • .

16. The form of the exact solution to is

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18. Euler’s Method

19. Euler’s method of solving ordinary differential equations states

20. To solve the ordinary differential equation by Euler’s method, you need to rewrite the equation as • .

21. The order of accuracy for a single step in Euler’s method is • O(h) • O(h2) • O(h3) • O(h4)

22. The order of accuracy from initial point to final point while using more than one step in Euler’s method is • O(h) • O(h2) • O(h3) • O(h4)

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24. Do you know how Runge- Kutta 4th Order Method works? • Yes • No • Maybe • I take the 5th

25. RUNGE-KUTTA 4TH ORDER METHOD

26. Runge-Kutta 4th Order Method

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28. Physical Examples

29. Ordinary Differential Equations Problem: The trunnion initially at room temperature is put in a bath of dry-ice/alcohol. How long do I need to keep it in the bath to get maximum contraction (“within reason”)?

30. Assumptions The trunnion is a lumped mass system. • What does a lumped system mean? It implies that the internal conduction in the trunnion is large enough that the temperature throughout the ball is uniform. • This allows us to make the assumption that the temperature is only a function of time and not of the location in the trunnion.

31. Energy Conservation Heat In – Heat Lost = Heat Stored

32. Heat Lost Rate of heat lost due to convection= hA(T-Ta) h = convection coefficient (W/(m2.K)) A = surface area, m2 T= temp of trunnion at a given time, K

33. Heat Stored Heat stored by mass = mCT where m = mass of ball, kg C = specific heat of the ball, J/(kg-K)

34. Energy Conservation Rate at which heat is gained – Rate at which heat is lost =Rate at which heat is stored 0- hA(T-Ta) = d/dt(mCT) 0- hA(T-Ta) = m C dT/dt

35. Putting in The Numbers Length of cylinder = 0.625 m Radius of cylinder = 0.3 m Density of cylinder material  = 7800 kg/m3 Specific heat, C = 450 J/(kg-C) Convection coefficient, h= 90 W/(m2-C) Initial temperature of the trunnion, T(0)= 27oC Temperature of dry-ice/alcohol, Ta = -78oC

36. The Differential Equation Surface area of the trunnion A = 2rL+2r2 = 2**0.3*0.625+2**0.32 = 1.744 m2 Mass of the trunnion M =  V =  (r2L) = (7800)*[*(0.3)2*0.625] = 1378 kg

37. The Differential Equation

38. Solution Time Temp (s) (oC) 0 27 1000 0.42 2000 -19.42 3000 -34.25 4000 -45.32 5000 -53.59 6000 -59.77 7000 -64.38 8000 -67.83 9000 -70.40 10000 -72.32

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40. If assigned HW every class for a grade, you predict that you would get a • better overall grade • same overall grade (would not make a difference) • lower overall grade

41. If I had given you a choice of taking the class online or in-class, and class attendance was not mandatory for in-class section, what would have been your choice? (you would have the same graded assignments and had to come to campus to take the tests for either section) • In-class • Online

42. If I had given you a choice of taking the class online or in-class but required 80% attendance for in-class section, what would have been your choice? (you would have the same graded assignments and had to come to campus to take the tests for either section) • In-class • Online

43. How likely are you to watch the YouTube videos for the topics that were presented in the class you missed? • Certainly • Likely • Not likely • Not at all

44. How likely are you to watch the YouTube videos for the topics that were presented in the class you attended? • Certainly • Likely • Not likely • Not at all

45. If based on your background such as learning patterns, GPA, etc, you were recommended to register for the online section or in-class section, how likely are you going to accept the recommendation? • Certainly • Likely • Not likely • Not at all

46. Given The value of at y(4) using finite difference method and a step size of h=4 can be approximated by • .