Introduction to Difference

Equations

Mathematics for Economists

Economic Dynamics

Economic dynamics is a study of how

economic variables evolve over time

Examples

Price change over time

GDP per capita change over time

Are examples of economic variables that change

over time

Definition of Difference Equation

A difference equation is an equation of the

change in a variable over discrete time

Example

is a difference equation

1 ttt yyy

Examples of Difference Equations

1032.9

32.8

3)log(2.7

32.6

23.5

132.4

433.3

33.2

2.1

123

2

1

1

2

1

1

21

12

1

1

tttt

tt

tt

tt

tt

ttt

ttt

tt

tt

yyyy

tyy

yy

yy

tyy

yyy

yyy

yy

yy

Classification of difference equations

by order Equations 1-9 above are all examples of

difference equations however these

equations have different characteristics

1. Some equations show change relationship

of two consecutive values y(t+1) and y(t) or

y(t) and y(t-1) called first order difference

equations Example

2

1

1

1

1

324

3)log(2.3

33.2

2.1

tyy

yy

yy

yy

tt

tt

tt

tt

Classification of difference equations

by order 2. Some equations show change relationships

between three consecutive values y(t+2) ,y(t+1)

and y(t) or y(t) , y(t-1)and y(t-2) called second

order difference equation

Example

tyyy

yyy

yyy

ttt

ttt

ttt

3)log(32.3

132.2

433.1

21

21

2

12

Classification of difference equations

by order 3. Some equations show change relationships

between more than three consecutive values

y(t+3), y(t+2) ,y(t+1) and y(t) called higher

order difference equations

Examples

1032.3

33.2

2.1

123

2

3

3

tttt

tt

tt

yyyy

yy

yy

Linear and Non-linear Difference

Equations Difference equations can also be classified as

linear or non-linear

A difference equation is nonlinear if it involves

any non-linear terms of it is

linear if all of the y terms are raised to no power

other than one.

etcyyy ttt ,,, 21

Examples of linear and non-linear

difference equations

1032.9

32.8

3)log(2.7

32.6

23.5

132.4

433.3

33.2

2.1

123

2

1

1

2

1

1

21

12

1

1

tttt

tt

tt

tt

tt

ttt

ttt

tt

tt

yyyy

tyy

yy

yy

tyy

yyy

yyy

yy

yy

Difference

equations

6,7and 8 are

linear while

others are non-

linear

Autonomous and Non-autonomous

Difference Equations Autonomous and Non-autonomous

Difference equations

A difference equation is said to be

autonomous if it does not depend on time

explicitly ;otherwise it is non-autonomous

Autonomous and Non-autonomous

difference equations

1032.9

32.8

3)log(2.7

32.6

23.5

132.4

433.3

33.2

2.1

123

2

1

1

2

1

1

21

12

1

1

tttt

tt

tt

tt

tt

ttt

ttt

tt

tt

yyyy

tyy

yy

yy

tyy

yyy

yyy

yy

yy Difference

equations 5and

8 are non-

autonomous

and others are

autonomous

difference

equations

Solutions of Difference Equations

The solution of a difference equation is a

function that satisfies the difference equation

true

Example: is solution of

The solution of a difference equation is not

unique as a result

t

ty 2

,...3,2,1,21 tyy tt

t

t Cy 2

Exercise

Classify each of the following according to

order, linear or non-linear , autonomous or

non-autonomous and difference or differential

equations

1032.9

32.8

3)log(2.7

32.6

23.5

2.4

2.3

12.2

0.1

123

2

1

1

2

1

1

2

2

3

12

1

1

tttt

tt

tt

tt

tt

tt

t

ttt

tt

yyyy

tyy

yy

yy

tyy

yy

tyy

yyy

yy

t

Solution Methods of first order linear

difference equations with constant

coefficients Definition: The general form of the linear

,first-order, autonomous difference equation

is given by :

,...2,1,0,1 tbayy tt

constants& knownarebawhere

Solution of First Order Linear

Autonomous Difference Equations Solving a difference equation is generally

finding the function of time that that

satisfies the difference equation ty

Solution Methods of first order linear

difference equations with constant

coefficients The linear ,first-order, autonomous difference

equation is given by :

is initial value first order autonomous

difference equation

,...2,1,0,1 tbayy tt

0ygiven and constants& knownarebawhere

Solution of initial value first order

linear difference equations with

constant coefficients Given initial value linear ,first-order,

autonomous difference equation :

solution is given by

,...2,1,0,1 tbayy tt

0ygiven and constants& knownarebawhere

1

,...2,1,0,11

1

0

0

aifbty

taifa

abya

y

tt

t

Solution of general first order linear

difference equations with constant

coefficients Given the general linear ,first-order,

autonomous difference equation :

solution is given by

,...2,1,0,1 tbayy ttconstants& knownarebawhere

1

,...2,1,0,11

1

aifbtC

taifa

abCa

y

tt

t

Solve each of the following initial value first order

linear difference equations with constant

coefficients

2,123.

50,505.0.

1,13.

2,2.

01

01

01

01

yyyd

yyyc

yyyb

yyya

tt

tt

tt

tt

Steady state solution

Steady State or stationary value in a

linear first order autonomous

difference equation is defined as the

value of y at which the system comes

to rest. That is the case where

tt yy 1

Steady state solution

Do the price of fuel increase indefinitely or

will converge to some number at some point

in time?

Steady state solution

Given the general linear ,first-order,

autonomous difference equation :

The steady state solution is given by

,...2,1,0,1 tbayy ttconstants& knownarebawhere

1,1

aa

by

Exercises For each of the following difference equations

Solve for the steady state if it exists and indicate

whether or not converge to the steady state

ty

2,123.

50,505.0.

1,13.

2,2.

01

01

01

01

yyyd

yyyc

yyyb

yyya

tt

tt

tt

tt

Application

Simple interest and compound interest

Introduction to Differential

Equations

5/25/2015 26 Mathematics for Economists

Definition:

A differential equation is an equation containing an unknown function

and its derivatives.

32 xdx

dy

032

2

aydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

Examples:.

y is dependent variable and x is independent variable, and these are ordinary

differential equations since only ordinary derivatives are involved in the equation

1.

2.

3.

ordinary differential equations

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Partial Differential Equation

Examples:

02

2

2

2

y

u

x

u

04

4

4

4

t

u

x

u

t

u

t

u

x

u

2

2

2

2

u is dependent variable and x and y are independent variables,

and is partial differential equation.

u is dependent variable and x and t are independent variables

1.

2.

3.

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Order of Differential Equation

The order of the differential equation is order of the highest

derivative in the differential equation.

Differential Equation ORDER

32 xdx

dy

0932

2

ydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

1

2

3

5/25/2015 29 Mathematics for Economists

Degree of Differential Equation

Differential Equation Degree

03

2

2

aydx

dy

dx

yd

36

4

3

3

y

dx

dy

dx

yd

03

53

2

2

dx

dy

dx

yd

1

1

3

The degree of a differential equation is power of the highest

order derivative term in the differential equation.

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Linear Differential Equation

A differential equation is linear, if

1. dependent variable and its derivatives are of degree one,

2. coefficients of a term does not depend upon dependent

variable.

Example:

36

4

3

3

y

dx

dy

dx

yd

is non - linear because in 2nd term is not of degree one.

.0932

2

ydx

dy

dx

ydExample:

is linear.

1.

2.

5/25/2015 31 Mathematics for Economists

Example: 3

2

22 x

dx

dyy

dx

ydx

is non - linear because in 2nd term coefficient depends on y.

3.

Example:

is non - linear because

ydx

dysin

!3

sin3y

yy is non linear

4.

5/25/2015 32 Mathematics for Economists

It is Ordinary/partial Differential equation of order and of degree, it is linear / non linear, with independent variable, and dependent variable.

5/25/2015 33 Mathematics for Economists

Autonomous ordinary differential equation

Mostly economic dynamics involves time as independent variable and differential

equations of time as independent variable

are common in economic modeling

A differential equation of time is said to be autonomous if it does not depend on time

explicitly otherwise it is non-autonomous

5/25/2015 Mathematics for Economists 34

Examples

5/25/2015 Mathematics for Economists 35

0.4

cos.3

35.2

5.1

2

2

yy

tyty

yy

tyy Differential equations 2 and 4

are autonomous

and 1 and 3 are

non-autonomous

differential

equations

Notations: dt

dyy

1st order differential equation

2. Differential form:

01 ydxdyx.

.0),,( dx

dyyxf),( yxf

dx

dy

3. General form:

or

1. Derivative form:

xgyxadx

dyxa 01

36 Mathematics for Economists 5/25/2015

First Order Ordinary Differential

equation

37 Mathematics for Economists 5/25/2015

Second order Ordinary Differential

Equation

38 Mathematics for Economists 5/25/2015

nth order linear differential equation

1. nth order linear differential equation with constant coefficients.

xgyadx

dya

dx

yda

dx

yda

dx

yda

n

n

nn

n

n

012

2

21

1

1 ....

2. nth order linear differential equation with variable coefficients

xgyxadx

dyxa

dx

ydxa

dx

ydxa

dx

dyxa

n

n

nn

012

2

2

1

1 ......

39 Mathematics for Economists 5/25/2015

Autonomous Equations

Definition: The general form of the linear autonomous first order differential

equation is

where are constants

5/25/2015 Mathematics for Economists 40

bayy

ba &

Examples

Each of the following are autonomous first order linear differential equations

5/25/2015 Mathematics for Economists 41

0.4

03.0.3

35.2

25.1

yy

yy

yy

yy

Solution Methods

For different types of differential equations there are different methods of solving but

generally the solution of a differential

equation is a function that satisfies the

differential equation

For example for , is a solution

5/25/2015 Mathematics for Economists 42

02

yytey 2

Solution methods of autonomous

differential equations

Given where are

constants

If b=0, the equation is called

homogeneous otherwise it is called non-

homogeneous differential equation

5/25/2015 Mathematics for Economists 43

bayy

ba &

0

ayy

Examples

Equations 3 & 4 are homogeneous while 1 and 2 are non-homogeneous

5/25/2015 Mathematics for Economists 44

0.4

03.0.3

35.2

25.1

yy

yy

yy

yy

Solution methods of autonomous differential equations

Given where are constants

Solving it involves two steps

1. Finding the solution of the complementary

homogeneous equation

2. Finding the a particular solution where

3. And finally the general solution is given by

5/25/2015 Mathematics for Economists 45

bayy

ba &

hyayy ,0

0

y

ph yyy

Theorem( Homogeneous solution)

The general solution of the homogeneous form of the linear , autonomous , first order

differential equation is where is

a constant

5/25/2015 Mathematics for Economists 46

0

ayy

atCeyh

a

Exercise

Solve each of the following differential equations

5/25/2015 Mathematics for Economists 47

0.4

03.0.3

05.2

05.1

yy

yy

yy

yy

Particular Solution

A steady state solution of a differential equation is the solution of the differential

equation where hence the

steady state solution of the where

are constants is , this steady

state solution is particular solution of

Hence the general solution of is

5/25/2015 Mathematics for Economists 48

0

dt

dyy

bayy

ba &a

by

bayy

bayy

a

bCeyyy athp

Examples

Find the general solution of each of the following differential equations

5/25/2015 Mathematics for Economists 49

0.4

23.0.3

35.2

25.0.1

yy

yy

yy

yy

The Initial Value Problem (IVP)

Given where are constants and if initial value is

given the problem is called initial value

problem and the general solution will give

rise to the solution

5/25/2015 Mathematics for Economists 50

bayy

ba &

0)0( yy

a

be

a

byyyy athp

0

Exercises: Solve the following

differential equations

5/25/2015 Mathematics for Economists 51

15.

10122/12.

10123.

10.

0

0

0

0

yandyd

yandyyc

yandyyb

yandyya

5/25/2015 Mathematics for Economists 52

342.

512.

27.

10844.

1510.

366.

0

0

0

0

0

0

yandyya

yandyyi

yandyh

yandyyg

yandyyf

yandyye

Examples

Solve each of the following IVP

5/25/2015 Mathematics for Economists 53

4/1,05

1.4

3,23.0.3

2,35.2

10,25.0.1

0

0

0

0

yyy

yyy

yyy

yyy

Steady Sate and convergence

Given where are constants the solution is converges to the

steady state solution no matter what

the initial value if and only if the coefficient

of the differential equation

5/25/2015 Mathematics for Economists 54

bayy

ba &

a

by

0a

Example

Discuss the convergence of the solution of each of the following ODEs

5/25/2015 Mathematics for Economists 55

4/1,05

1.4

3,23.0.3

2,35.2

10,25.0.1

0

0

0

0

yyy

yyy

yyy

yyy