1. Solutions of Second Order Linear ODEs The Wronskian

2. Midterm Details Midterm on WEDNESDAY, in class Covers all sections up to 3.2 EXCEPT 2.4 Bring a Blue Book 1 Page (front and back) of Handwritten notes 50 minutes, 5 questions, 50 points

3. Solutions of Linear ODEs Remember from Friday:

4. Solutions of Linear ODEs Remember from Friday: Has general solutions: and

5. Solutions of Linear ODEs Remember from Friday: Has general solutions: and Which leads to the general form of solution:

6. Solutions of Linear ODEs Remember from Friday: Has general solutions: and Which leads to the general form of solution: Or:

7. Solutions of Linear ODEs Remember from Friday: Has general solutions: and Which leads to the general form of solution: Or:

8. Solutions of Linear ODEs Remember from Friday: Has general solutions: and Which leads to the general form of solution: Or:

9. Solutions of Linear ODEs Remember from Friday: Has general solutions: and Which leads to the general form of solution: Or:

10. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ?

11. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ?

12. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ?

13. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ?

14. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ?

15. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

16. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

17. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

18. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

19. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

20. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

21. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

22. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

23. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

24. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

25. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

26. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

27. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

28. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

29. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

30. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, Why ? So …

31. Solutions of Linear ODEs This holds for general linear ODEs If Then so is and are solutions, In fact, so is

32. But that’s good! Two constants allow us to fit initial conditions By solving the equations But can we always findand? In other words, can we always solve the system of equations?

33. The Wronskian Can we always solve forand? Recall the matrix formulation: And the determinant If the determinant of a matrix is non-zero, there is a unique solution to the system of equations

34. The Wronskian Can we always solve forand? Take the determinant If the determinant is NOT zero Can always findand

35. The Wronskian What about different starting times? Take the determinant If the determinant is NOT zero Can always findand

36. The Wronskian What about different starting times? Take the determinant If the determinant is NOT zero Can always findand

37. The Wronskian Take the determinant If the determinant is NOT zero Can always findand What about different starting times?

38. The Wronskian Take the determinant If the determinant is NOT zero Can always findand In general: Is called the Wronskian function.

39. Fundamental Set of Solutions For two solutions of an ODE If the Wronskian is non- zero The solutions are said to form a Fundamental Set of Solutions Because we can fit any initial conditions

40. Fundamental Set of Solutions To verify Form a fundamental set of solutions to Verify that Compute the Wronskian and satisfy the ODE Verify the Wronskian is not 0

41. Example Show : Form a fundamental set of solutions to Verify solutions:

42. Example Show : Form a fundamental set of solutions to (Repeat for ) Compute Wronskian:

43. Example Show : Form a fundamental set of solutions to (Repeat for ) Compute Wronskian: Verify the Wronskian is not 0 But what if the Wronskian can become 0?

44. Abel’s Theorem But what if the Wronskian can become 0? Abel’s (Amazing) Theorem: For linear homogeneous second order ODES: The Wronskian for a set of solutions is either ALWAYS ZERO or NEVER ZERO in the interval where the solution is exists.

45. Interval Where Solution Exists So how do we know interval where solution exists? Rewrite as:

46. Interval Where Solution Exists So how do we know interval where solution exists? Rewrite as:

47. Interval Where Solution Exists With initial conditions For : A unique, second differentiable solution exists on any open interval that contains AND where and are continuous.

48. Example What is the longest interval where has a unique solution?

49. Example What is the longest interval where has a unique solution? Rewrite as: Discontinuity at x=3 Discontinuity at x=0 and x=3 No discontinuity

50. Example What is the longest interval where has a unique solution? Rewrite as: Discontinuity at x=0 and x=3 Must contain x = 1

51. Example What is the longest interval where has a unique solution? Rewrite as: Longest interval is or

52. Summary Linear combinations of solutions to homogeneous linear ordinary differential equations are also solutions. The Wronskian determines where solutions form fundamental sets of equations. Existence and uniqueness are established by where the coefficient functions are continuous.

53. Questions?