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 solution of its reduced homogeneous equation

25. A solution of the Non-homogeneous Equation §A good technique to use to find a solution of a non-homogeneous equation is to try a linear combination of a 0 (t) and its first and second derivatives. Mathematics for Economics Beatrice Venturi 25

26. A solution of the Non-homogeneous Equation §If, a 0 (t) = 6t 3 -3t, then try to find values of A, B, C and D such that A + Bt + Ct 2 + Dt 3 is a solution. Mathematics for Economics Beatrice Venturi 26

27. The solution of the Non-homogeneous Equation §Or if §f (t) = 2sin t + cos t, then try to find values of A and B such that § f (t) = Asin t + Bcos t is a solution. §Or if § f (t) = 2e Bt §for some value of B, then try to find a value of A such that Ae Bt is a solution. Mathematics for Economics Beatrice Venturi 27

28. A solution of the Non-homogeneous Equation Mathematics for Economics Beatrice Venturi 28 The function on the right-hand side is a third-degree polynomial, so to find a solution of the equation, we have to try a general third-degree polynomial, that is, a function of the form: =A + Bt + Ct 2 + Dt 3. We consider for example :

29. Mathematics for Economics Beatrice Venturi 29 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

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 § The particular solution is: Thus the General solution of the original equation is

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

33. Mathematics for Economics Beatrice Venturi 33 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS x(t)=

34. Mathematics for Economics Beatrice Venturi 34 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

35. Mathematics for Economics Beatrice Venturi 35 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

36. Mathematics for Economics Beatrice Venturi 36 CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS