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Math 334 Lecture #7
§
2.5: Autonomous First Order ODE’s
Geometric Analysis of
y
0
=
f
(
y
).
The zeros of
f
are called the
critical points
of
the ODE.
The critical points correspond to the equilibrium solutions of the ODE.
Between consecutive zeros of
f
, the values of
f
are of one sign, so that
y
0
=
f
(
y
) are also.
Between consecutive zeros of
f
, solutions are increasing if
y
0
=
f
(
y
)
>
0 there, or they
are decreasing if
y
0
=
f
(
y
)
<
0 there.
A graph of the function
f
reveals the location of the zeros of
f
and the intervals where
f
is positive or negative.
Here is the graph of a function
f
for the purpose of this discussion:
5
2
!
1
3
f(y)
1
!
1
!
3
1
4
6
2
3
!
2
!
3
y
4
0
0
!
2
Note the zeros of
f
, and the sign of
f
between consecutive zeros.
The graph of
f
determines the phase portrait of the ODE
y
0
=
f
(
y
):
2
!
3
4
2$0
5
0$5
3$0
!
1
1$5
!
2
1$0
0
0$0
6
2$5
1
3
An equilibrium is
asymptotically stable
if every “nearby” solution converges to it as
t
→ ∞
.
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View Full Document[Identify on the phase portrait all of the equilibria that are asymptotically stable.]
[Question: Do any of the solutions near an asymptotically stable equilibrium
y
=
A
achieve the value of
A
in ﬁnite time? NO! Because if any did, it would contradict the
uniqueness of solutions.]
An equilibrium solution is
unstable
if every “nearby” solution moves away from it.
[Identify all of the unstable equilibria on the phase portrait.]
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 Spring '11
 Smith
 Critical Point

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