Unformatted text preview: CHAPTER 2 Continuous dynamic systems 2.1 Some deﬁnitions
A differential equation is an equation relating:
(a)
(b)
(c)
(d) the derivatives of an unknown function,
the function itself,
the variables in terms of which the function is deﬁned, and
constants. More brieﬂy, a differential equation is an equation that relates an unknown function
and any of its derivatives. Thus
dy
+ 3xy = ex
dx
is a differential equation. In general
dy
= f (x, y)
dx
is a general form of a differential equation.
In this chapter we shall consider continuous dynamic systems of a single variable.
In other words, we assume a variable x is a continuous function of time, t. A
differential equation for a dynamic equation is a relationship between a function
of time and its derivatives. One typical general form of a differential equation is
dx
= f (t, x)
dt (2.1) Examples of differential equations are:
(i) dx
+ 3x = 4 + e−t
dt (ii) d2 x
dx
+ 4t − 3(1 − t2 )x = 0
dt2
dt (iii) dx
= kx
dt (iv) ∂u ∂v
+
+ 4u = 0
∂t
dt ...
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 Fall '11
 Dr.Gwartney
 Economics

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