Economics Faculty

Mathematics for Economics Beatrice Venturi

1

Economics Faculty

CONTINUOUS TIME:LINEAR DIFFERENTIAL EQUATIONS

Economic Applications

LESSON 2prof. Beatrice Venturi


2

CONTINUOUS TIME : LINEAR ORDINARY

DIFFERENTIAL EQUATIONS

ECONOMIC APPLICATIONS


3

LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS (E.D.O.)

)1()()(' 01 xayxay Where f(x) is not a constant.In this case the solution has the form:

cdxxaeey

dxadxxa)(0

11 )(


4


dxxae )(1

)(

)()(

0

)(

1

)()(

1

11

xae

xyxaedx

dye

dxxa

dxxadxxa

We use the method of integrating factor and multiply by the factor:


5


)())(( 0

)()( 11 xaexyeDdxxadxxa

dxxaedxxyeDdxxadxxa

)())(( 0

)()( 11


6


GENERAL SOLUTION OF (1)

])([)( 0

)()( 11 cdxxaeexydxxadxxa


7

FIRST-ORDER LINEAR E. D. O.

)(xxydx

dy

xdxxy

dy

)(

2

2

)(x

cexy

Example


8


y′-xy=0

y(0)=1

We consider the solution when we assign an initial condition:



9

2

2

)0()(x

eyxy

When any particular value is substituted for C; the solution became a particular solution:

2

2

)(x

Cexy

The y(0) is the only value that can make the solution satisfy the initial condition. In our case y(0)=1

2

2

)(x

exy


10


[Plot]

52.50-2.5-5

2.5e+5

2e+5

1.5e+5

1e+5

5e+4

0

x

y

x

y

2

2

)(x

exy


11

The Domar Model

)(1

)(

1ts

Idt

dII

tsdt

dI


12

The Domar Model

Where s(t) is a t function

0)( Itsdt

dI

dttsCetI )()(


13

LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS

The homogeneous case:

0)()()1( 1 xyxadx

dy


14


).()(1 xyxadx

dy

dxxaxy

dy)(

)( 1

dxxaxy )()(ln 1

dxxaCexy )(1)(

Separate variable the to variable y and x:

We get:


15


We should able to write the solution of (1).

solutionGeneral

dxxaCety )(1)(


16


2) Non homogeneous Case :

)()()()2( 01 xaxyxadx

dy


17

CONTINUOUS TIME: SECOND ORDER DIFFERENTIAL EQUATIONS

We have two cases:

• homogeneous;

• non omogeneous.


18


)()( 0122

2

tatxadt

dxa

dt

xd

: a)Non homogeneous case with constant coefficients

b)Homogeneous case with constant coefficients

0)(122

2

txadt

dxa

dt

xd


19


tCetx )(

tt edt

xdande

dt

dx 22

2

We adopt the trial solution:


20


We get:

0)( 122 aae t

This equation is known as characteristic equation

0122 aa


21


Case a) : We have two different roots

21 andThe complentary function:

the general solution of its reduced homogeneous equation istt ecectx 21

21)( where Rcandc 21

are two arbitrary function.


22


Caso b) We have two equal roots

21

tt tecectx 21)(

dove 21 cec

sono due costanti arbitrarie

The complentary function:the general solution of its reduced homogeneous equation is


23

Case c) We have two complex conjugate roots

i1

, i2

The complentary function:the general solution of its reduced homogeneous

equation istektektx tt sincos)( 21

This expession came from the Eulero Theorem



24


Examples

tttxdt

dx

dt

xd3)(32 3

2

2

0322

311)

2(

22

2/1 a

acbb

3,1 21

The complentary function:The solution of its reduced homogeneous equation

tttt eetxeetx 321

21 )(,)(

tt ecectx 321)(


25


dctbtattx 23)(

cbtattx 23)(' 2

battx 26)(''


26


ttdctbtat

cbtatbat

3)(3

)23(226323

2

0322

03346

036

013

dcb

cba

ba

a


27


The particular solution::

27

22

9

5

3

2

3

1)( 23 ttttx

)()( 321 txtecectx tt

The General solution


28


The Cauchy Problem

1)0( x

0)0( x

tttxdt

dx

dt

xd3)(32 3

2

2


29


23t2 5

9t 1

3t3 e t 5

27e3t 22

27x(t)=

52.50-2.5-5

5e+5

3.75e+5

2.5e+5

1.25e+5

0

x

y

x

y


30


21

tt tecectx 21)(

0)(91242

2

txdt

dx

dt

xd

2

3

4

36)6(6 2

2/1

09124 2


31


0)(522

2

txdt

dx

dt

xd

i211 i212

)2sin2(cos)(1 titetx t

)2sin2(cos)(2 titetx t


32


tetxtx

t t 2cos2

)()()( 21

1

tei

txtxt t 2sin

2

)()()( 21

2

tektektx tt 2sin2cos)( 21

Documents

Economics Faculty