Download presentation

Presentation is loading. Please wait.

Published byColby Tillis 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 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

Similar presentations

OK

GG Consulting, LLC I-SUITE. Source: TEA SHARS Frequently asked questions 2.

GG Consulting, LLC I-SUITE. Source: TEA SHARS Frequently asked questions 2.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Convert word doc to ppt online training Ppt on file system in unix everything is a file Ppt on solid dielectrics effect Ppt on series and parallel combination of resistors in series Ppt on video teleconferencing Png to ppt online converter Ppt on colonialism and tribal societies Free ppt on etiquettes of life Ppt online shopping cart system Best ppt on save water