1. Mathe III Lecture 5 Mathe III Lecture 5 Mathe III Lecture 5 Mathe III Lecture 5

2. 2 Stability: In the long run, the solution should be independent of the initial conditions. The general solution of is: if : The system is stable.

3. 3 if 1 m The root (s) are in (-1, 1) iff:

4. 4 1 m The system is stable iff:

5. 5 Differential Equations First Order Differential Equations first order, ordinary equation (single variable) Differential Equations

6. 6 x t

7. 7 The simplest possible equation: x t

8. 8 An approximation: For a given let: we obtain a difference equation, solve it and let or graphically:

9. 9 x For t = 0, assume x(0) = x 0 x0x0 t x1x1 x2x2 etc.

10. 10 x For t = 0, assume x(0) = x 0 x0x0 t x1x1 x2x2 Now choose a smaller As we approach a curve which solves

11. 11

12. 12 Separable Differential Equations A formal ‘trick’:

13. 13 Is this ‘trick’ valid ???

14. 14 This defines x as an implicit function of t

15. 15

16. 16

17. 17

18. 18 Separable Differential Equations (again)

19. 19 Separable Differential Equations (again)

20. 20 Graphic description of the solution

21. 21 Graphic description of the solution

22. 22 t x Graphic description of the solution

23. 23 t x Graphic description of the solution

24. 24 t x Graphic description of the solution

25. 25

26. 26

27. 27 This enables us to study how the evolution of capital changes with the parameters

28. 28 How does K/L behave in the long run?

29. 29 How does K/L behave in the long run?