Ordinary Differential Equations Everything is ordinary about them

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?

5. END

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

17. END

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)

23. END

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

27. END

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

39. END

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 • .