Exercises 5.4
figure indicates that (0
,
0) is a stable critical point (center) whereas (1
,
0) is a saddle point
(unstable).
25.
This system has two critical points, (0
,
0) and (1
,
0), which are solutions to the system
y
= 0
,
−
x
+
x
3
= 0
.
The direction field for this system is depicted in Figure B.37. From this figure we conclude
that
(a)
the solution passing through the point (0
.
25
,
0
.
25) ﬂows around (0
,
0) and thus is periodic;
(b)
for the solution (
x
(
t
)
, y
(
t
)) passing through the point (2
,
2),
y
(
t
)
→ ∞
as
t
→ ∞
, and so
this solution is not periodic;
(c)
the solution passing through the critical point (1
,
0) is a constant (equilibrium) solution
and so is periodic.
27.
The direction field for given system is shown in Figure B.38 in the answers of the text. From
the starting point, (1
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Critical Point, Rightwing politics, Leftwing politics, Euclidean geometry, saddle point, direction ﬁeld

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