Systems of Frstorder differential equations
151
Figure 4.6.
Consider the frst equilibrium line. Along this line we have combinations oF
x
and
y
For which ˙
x
=
0. But this means that For any value oF
x
on this line,
x
cannot be
changing. This inFormation is shown by the vertical dotted lines in fgure 4.6 For
any particular value oF
x
. Similarly, For the line denoted
˙
y
=
0(
y
=
2
x
)
the value oF
y
on this line cannot be changing. This inFormation is shown by the
horizontal dotted lines in fgure 4.6 For any particular value oF
y
.
Next consider points either side oF the equilibrium lines in the phase plane. To
the right oF the
x
line we have
y
<
x
3
implying
˙
x
>
0
Hence, For any point at which
x
lies below the
x
line, then
x
is rising. Two are
shown in fgure 4.6 by the
horizontal
arrows that are pointing to the right. By the
same reasoning, to the leFt oF the
x
line we have
y
>
x
3
implying
˙
x
<
0
Hence, For any point at which
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 Fall '11
 Dr.Gwartney
 Economics, Rightwing politics, Leftwing politics, firstorder differential equations, Horizontal arrows, ﬁrst equilibrium line

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