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Mathe III Lecture 5

Mathe III Lecture 5. Mathe III Lecture 5. Mathe III Lecture 5. Stability:. In the long run, the solution should be independent of the initial conditions. The general solution of. is:. if :. The system is stable. if. The root (s) are in (-1, 1) iff:. m. -1. 1. m. -1. 1.

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Mathe III Lecture 5

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

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

  4. m -1 1 The system is stable iff:

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

  6. x t

  7. The simplest possible equation: x t

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

  9. For t = 0, assume x(0) = x0 x x2 x1 etc. x0 t

  10. For t = 0, assume x(0) = x0 Now choose a smaller x x2 x1 As we approach a curve which solves x0 t

  11. Separable Differential Equations A formal ‘trick’:

  12. Is this ‘trick’ valid ???

  13. This defines x as an implicit function of t

  14. Separable Differential Equations (again)

  15. Separable Differential Equations (again)

  16. Graphic description of the solution

  17. Graphic description of the solution

  18. Graphic description of the solution x t

  19. Graphic description of the solution x t

  20. Graphic description of the solution x t

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

  22. How doesK/L behave in the long run?

  23. How doesK/L behave in the long run?

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