Fall 2003 Math 308/501–502
1 Introduction
1.D Autonomous Equations, Stability,
and the Phase Line
Fri, 05/Sep
c 2003, Art Belmonte
Summary
Autonomous Equations
A firstorder
autonomous
differential
equation is one of the form
dy
/
dt
=
y
0
=
f
(
y
)
. Notice that the
derivative expression does not depend on the independent variable
t
. Accordingly, along a given horizontal line in the
ty
plane (say
y
=
b
), the slopes are the same. Moreover, tangent line segments
in the direction field along the vertical line
t
=
a
are replicated
along any other vertical line
t
=
a
+
k
. Not only does this give the
direction field a rather uniform appearance, it also means that the
DE lends itself readily to
qualitative
analysis of solution curves.
That is, without even analytically solving the DE (if this is even
possible), we can still say a lot about the behavior of its solutions.
Equilibria
To do this, first find the zeros of
f
; i.e., the values
y
for which
f
(
y
)
=
0. If
f
(
c
)
=
0, then the
constant
function
y
≡
c
is a solution of the DE, since
y
0
≡
0
=
f
(
c
)
=
f
(
y
(
t
))
for
all
t
. The value
c
is called an
equilibrium point
and the constant
function
y
≡
c
is called an
equilibrium solution
. Looking at a
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 Spring '08
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 Math, Equations, Line segment, Equilibrium point, Stability theory, phase line

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