1. First Order Linear Equations Integrating Factors

2. First Order Linear Equations Only First Derivative Of the form: Notice, we can rewrite:

3. First Order Linear Equations Only First Derivative Of the form: Notice, we can rewrite: So we consider general form:

4. Reversing Differentiation Rules Friday: Chain Rule Separation of Variables

5. Reversing Differentiation Rules Friday: Chain Rule Separation of Variables This is nonlinear

6. Reversing Differentiation Rules Friday: Chain Rule Separation of Variables We need a different technique.

7. Reversing Differentiation Rules Friday: Chain Rule Separation of Variables Today: Product Rule Integrating Factors

8. Product Rule

9. Want to rewrite as Then:

10. Product Rule Want to rewrite as Then:

11. Product Rule Want to rewrite as Then: Solution!

12. The Catch… is not of form

13. The Catch… (If it was, then )

14. So we introduce an Integrating Factor

18. (Challenge - Derive this using separation of variables)

19. Why? (Chain Rule) (Fundamental Theorem of Calculus)

20. So To Solve… Compute Integrating Factor Multiply Through Rewrit e Integrate

21. So To Solve… Compute Integrating Factor Multiply Through Rewrit e Integrate

22. Example Compute Integrating Factor

23. Example

24. Multiply Through By Integrating Factor

25. Example Multiply Through By Integrating Factor

26. Example Multiply Through By Integrating Factor

27. Example Rewrite Left Hand Side

28. Example Rewrite Left Hand Side

29. Example Integrate Both Sides

30. Example

31. (u sub)

32. Example Implicit Explicit

33. Long Term Behavior Solve s Equilibrium Solution:

34. Long Term Behavior Solve s Equilibrium Solution:

35. Long Term Behavior Solve s Equilibrium Solution:

36. Long Term Behavior Solve s Equilibrium Solution: What happens to general solutions over long times? For Any Constant C

37. Long Term Behavior Solve s Equilibrium Solution: What happens to general solutions over long times? For Any Constant C So we say solutions “converge” to y = 5/3 This also tells us that y = 5/3 is stable.

38. Long Term Behavior

39. Summary First Order Linear Equation: Solve Using Integrating Factors: Long term behavior determined by taking limit as t goes to infinity

40. Questions?