CHAPTER 4
Systems of firstorder
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 firstorder 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 firstorder 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 firstorder differential
equations because of the term
xy
occurring on the righthand side of both equations
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 Fall '11
 Dr.Gwartney
 Economics, Derivative, economic problems, Nonlinear system

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