Economics Dynamics Problems 158 - CHAPTER 4 Systems of...

CHAPTER 4
Systems of first-order
differential equations
4.1 Definitions and autonomous systems
In many economic problems the models reduce down to two or more systems of
differential equations that require to be solved simultaneously. Since most eco-
nomic models reduce down to two such equations, and since only two variables
can easily be drawn, we shall concentrate very much on a system of two equations.
In general, a system of two ordinary first-order differential equations takes the form
dx
dt
= ˙ x = f ( x , y , t )
dy
dt
= ˙ y = g ( x , y , t )
(4.1)
Consider the following examples in which x and y are the dependent variables
and t is an independent variable:
( i )
˙ x = ax − by − ce
t
˙ y = rx + sy − qe
t
( ii )
˙ x = ax − by
˙ y = rx + sy
( iii )
˙
x = ax − bxy
˙ y = rx − sxy
Examples (i) and (ii) are linear systems of first-order differential equations
because they involve the dependent variables x and y in a linear fashion.
Example (iii), on the other hand, is a nonlinear system of first-order differential
equations because of the term xy occurring on the right-hand side of both equations