1. Mathematics for Economics Beatrice Venturi 1 Economics Faculty CONTINUOUS TIME: LINEAR DIFFERENTIAL EQUATIONS Economic Applications LESSON 2 prof. Beatrice Venturi

2. Mathematics for Economics Beatrice Venturi 2 CONTINUOUS TIME : LINEAR ORDINARY DIFFERENTIAL EQUATIONS ECONOMIC APPLICATIONS

3. 3 LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) Where f(x) is not a constant. In this case the solution has the form:

4. Mathematics for Economics Beatrice Venturi 4 LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) We use the method of integrating factor and multiply by the factor:

5. Mathematics for Economics Beatrice Venturi 5 LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.)

6. Mathematics for Economics Beatrice Venturi 6 LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) GENERAL SOLUTION OF (1)

7. Mathematics for Economics Beatrice Venturi 7 FIRST-ORDER LINEAR E. D. O. Example

8. Mathematics for Economics Beatrice Venturi 8 FIRST-ORDER LINEAR E. D. O. y-xy=0 y(0)=1 We consider the solution when we assign an initial condition :

9. FIRST-ORDER LINEAR E. D. O. Mathematics for Economics Beatrice Venturi 9 When any particular value is substituted for C; the solution became a particular solution: The y(0) is the only value that can make the solution satisfy the initial condition. In our case y(0)=1

10. Mathematics for Economics Beatrice Venturi 10 FIRST-ORDER LINEAR E. D. O. §[Plot]

11. Mathematics for Economics Beatrice Venturi 11 The Domar Model

12. Mathematics for Economics Beatrice Venturi 12 The Domar Model §Where s(t) is a t function

13. Mathematics for Economics Beatrice Venturi 13 LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS §The homogeneous case:

14. Mathematics for Economics Beatrice Venturi 14 LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS Separate variable the to variable y and x: We get:

15. Mathematics for Economics Beatrice Venturi 15 LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS We should able to write the solution of (1).

16. Mathematics for Economics Beatrice Venturi 16 LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.) 2) Non homogeneous Case :

17. Mathematics for Economics Beatrice Venturi 17 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS §We have two cases: homogeneous; non omogeneous.

18. Mathematics for Economics Beatrice Venturi 18 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS : a)Non homogeneous case with constant coefficients b)Homogeneous case with constant coefficients

19. Mathematics for Economics Beatrice Venturi 19 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS We adopt the trial solution:

20. Mathematics for Economics Beatrice Venturi 20 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS We get: This equation is known as characteristic equation

21. Mathematics for Economics Beatrice Venturi 21 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Case a) : We have two different roots The complentary function: the general solution of its reduced homogeneous equation is where are two arbitrary function.

22. Mathematics for Economics Beatrice Venturi 22 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS Caso b) We have two equal roots dove sono due costanti arbitrarie The complentary function: the general solution of its reduced homogeneous equation is

23. Mathematics for Economics Beatrice Venturi 23 Case c) We have two complex conjugate roots, The complentary function: the general solution of its reduced homogeneous equation is This expession came from the Eulero Theorem CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

24. Mathematics for Economics Beatrice Venturi 24 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS §Examples The complentary function: The solution of its reduced homogeneous equation

25. Mathematics for Economics Beatrice Venturi 25 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

26. Mathematics for Economics Beatrice Venturi 26 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

27. Mathematics for Economics Beatrice Venturi 27 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS § The particular solution:: The General solution

28. Mathematics for Economics Beatrice Venturi 28 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS The Cauchy Problem

29. Mathematics for Economics Beatrice Venturi 29 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS x(t)=

30. Mathematics for Economics Beatrice Venturi 30 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

31. Mathematics for Economics Beatrice Venturi 31 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

32. Mathematics for Economics Beatrice Venturi 32 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS