Mathe III Lecture 5 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 1 m The root (s) are in (-1, 1) iff:

4 1 m The system is stable iff:

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

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

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

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12 Separable Differential Equations A formal ‘trick’:

13 Is this ‘trick’ valid ???

14 This defines x as an implicit function of t

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18 Separable Differential Equations (again)

19 Separable Differential Equations (again)

20 Graphic description of the solution

21 Graphic description of the solution

22 t x Graphic description of the solution

23 t x Graphic description of the solution

24 t x Graphic description of the solution

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

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

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