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Section
5.
Graphing
Systems
5A.
The
Phase
Plane
5A1.
Find
the
critical
points
of
each
of
the
following
nonlinear
autonomous
systems.
x
�
=
x
2
−
y
2
x
=
1
−
x
+
y
a)
�
b)
�
y
=
x
−
xy
y
=
y
+
2
x
2
5A2.
Write
each
of
the
following
equations
as
an
equivalent
firstorder
system,
and
find
the
critical
points.
a)
x
��
+
a
(
x
2
−
1)
x
�
+
x
=
0
b)
x
��
−
x
�
+
1
−
x
2
=
0
5A3.
In
general,
what
can
you
say
about
the
relation
between
the
trajectories
and
the
critical
points
of
the
system
on
the
left
below,
and
those
of
the
two
systems
on
the
right?
x
=
f
(
x,
y
)
x
�
=
−
f
(
x,
y
)
x
=
g
(
x,
y
)
a)
y
=
g
(
x,
y
)
y
�
=
−
g
(
x,
y
)
b)
y
�
=
−
f
(
x,
y
)
5A4.
Consider
the
autonomous
system
x
=
f
(
x,
y
)
x
(
t
)
;
solution
:
x
=
.
y
=
g
(
x,
y
)
y
(
t
)
a)
Show
that
if
x
1
(
t
)
is
a
solution,
then
x
2
(
t
) =
x
1
(
t
−
t
0
)
is
also
a
solution.
What
is
the
geometric
relation
between
the
two
solutions?
b)
The
existence
and
uniqueness
theorem
for
the
system
says
that
if
f
and
g
are
contin
uously
differentiable
everywhere,
there
is
one
and
only
one
solution
x
(
t
)
satisfying
a
given
initial
condition
x
(
t
0
)
=
x
0
.
Using
this
and
part
(a),
show
that
two
trajectories
cannot
intersect
anywhere.
(Note
that
if
two
trajectories
intersect
at
a
point
(
a,
b
),
the
corresponding
solutions
x
(
t
)
which
trace
them
out
may
be
at
(
a,
b
)
at
different
times.
Part
(a)
shows
how
to
get
around
this
diﬃculty.)
5B.
Sketching
Linear
Systems
x
=
−
x
5B1.
Follow
the
Notes
(GS.2)
for
graphing
the
trajectories
of
the
system
�
y
=
−
2
y
.
dy
a)
Eliminate
t
to
get
one
ODE
=
F
(
x,
y
).
Solve
it
and
sketch
the
solution
curves.
dx
b)
Solve
the
original
system
(by
inspection,
or
using
eigenvalues
and
eigenvectors),
obtaining
the
equations
of
the
trajectories
in
parametric
form:
x
=
x
(
t
)
,
y
=
y
(
t
).
Using
these,
put
the
direction
of
motion
on
your
solution
curves
for
part
(a).
What
new
trajectories
are
there
which
were
not
included
in
the
curves
found
in
part
(a)?
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 Fall '09
 vogan
 Equations, Critical Point, Linear system, Nonlinear system, Stability theory, x�, Sketching Linear Systems

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