1. 1 Economic Faculty Differential Equations and Economic Applications LESSON 1 prof. Beatrice Venturi

2. Beatrice Venturi2 DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS

3. FIRST ORDER DIFFERENTIAL EQUATIONS DEFINITION: Let y(x) =“ unknown function” x = free variable y' = first derivative Beatrice Venturi 3 First order Ordinary Differential Equation.

4. FIRST ORDER DIFFERENTIAL EQUATIONS DEFINITION: An ordinary differential equation (or ODE) is an equation involving derivates of: y(x) (the unknown function) a real value function (of only one independent variable x) defined in y: (a,b)  R an open interval (a,b). Beatrice Venturi4

5. FIRST ORDERDIFFERENTIAL EQUATIONS More generally we may consider the following equation: Where f is the known function. Beatrice Venturi 5 (*)

6. Solution of E.D.O. Definition: A solution or integral curve of an EDO is a function g(x) such that when it is substituted into (*) it reduces (*) to an identity in a certain open interval (a,b) in R. We find a solution of an EDO by integration. Beatrice Venturi

7. 1.EXAMPLE Beatrice Venturi7

8. The Domar’s Growth Model Beatrice Venturi 8

9. Investment I and Capital Stock K Capital accumulation = process for which new shares of capital stock K are added to a previous stock. Beatrice Venturi9

10. Connection between Capital Stock and Investment Beatrice Venturi10 Capital stock= Investment =

11. Connection between Capital and Investment Beatrice Venturi11

12. Connection between Capital and Investment B eatrice Venturi12

13. Connection between Capital and Investment Beatrice Venturi13

14. Connection between Capital and Investment Beatrice Venturi14

15. Price adjustment in the market We consider the demand function : Beatrice Venturi15 and the supply function : for a commodity

16. Price adjustment in the market At the equilibrium when supply balances demand, the equilibrium prices satisfies: Beatrice Venturi16

17. Price adjustment in the market Beatrice Venturi 17 Suppose the market not in equilibrium initially. We study the way in which price varies over time in response to the inequalities between supply and demand.

18. Price adjustment in the market Beatrice Venturi18

19. Price adjustment in the market We use the method of integranting factors. We multiply by the factor Beatrice Venturi19

20. Price adjustment in the market Beatrice Venturi20 Solution = To find c put t=0

21. The equilibrium price P is asymptotically stable equilibrium Beatrice Venturi21

22. SEPARATION OF VARIABLES. This differential equation can be solved by separation of variables. Beatrice Venturi22 The method “ separates” the two variables y and x placing them in diffent sides of the equation:

23. Each sides is then integrated: Beatrice Venturi23

24. The Domar Model s(t)= marginal propensity to save is a function of t Beatrice Venturi24

25. PARTICULAR SOLUTION DEFINITION The particular integral or solution of E.D.O. Beatrice Venturi25 is a function : obtained by assigning particular values to the arbitrary constant

26. Example –Given the initial condition –the solution is unique Beatrice Venturi26

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28. Beatrice Venturi28 The graph of the particular solution

29. Case: C₁= 0 y=(1/3)x³ Beatrice Venturi29

30. Beatrice Venturi30 INTEGRALE SINGOLARE We have solution that cannot be obtained by assigning a value to a the constant c.

31. Beatrice Venturi31 Example:

32. Beatrice Venturi32 y=0 is a solution but this solution cannot be abtained by assing a value to c from the generale solution.