Download presentation

Presentation is loading. Please wait.

Published byBraxton Shirrell Modified over 3 years ago

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

Similar presentations

OK

©Brooks/Cole, 2001 Chapter 12 Derived Types-- Enumerated, Structure and Union.

©Brooks/Cole, 2001 Chapter 12 Derived Types-- Enumerated, Structure and Union.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on types of parallelograms pictures Ppt on review of literature examples Ppt on business etiquettes training day movie Ppt on application of trigonometry in different fields Ppt on asian continent countries Download ppt on 3d printing technology Ppt on paper display technology Ppt on pir sensor based security system Ppt on digital door lock system Download ppt on connectors in english grammar