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Unformatted text preview: CHAPTER 4 Systems of ﬁrst-order
differential equations 4.1 Deﬁnitions and autonomous systems (4.1) In many economic problems the models reduce down to two or more systems of
differential equations that require to be solved simultaneously. Since most economic 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
= x = f (x, y, t)
= y = g(x, y, t)
Consider the following examples in which x and y are the dependent variables
and t is an independent variable:
x = ax − by − cet
y = rx + sy − qet
x = ax − by
y = rx + sy
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
in the system. Examples (ii) and (iii) are autonomous systems since the variable
t does not appear explicitly in the system of equations; otherwise a system is said
to be nonautonomous, as in the case of example (i). Furthermore, examples (ii)
and (iii) are homogeneous because there is no additional constant. Example (i)
is nonhomogeneous with a variable term, namely cet .
A solution to system (4.1) is a pair of parametric equations x = x(t) and y =
y(t) which satisfy the system over some open interval. Since, by deﬁnition, they
satisfy the differential equation system, then it follows that the solution functions
are differentiable functions of t. As with single differential equations, it is often
necessary to impose initial conditions on system (4.1), which take the form
x0 = x(t0 ) and y0 = y(t0 ) ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.
- Fall '11