�
±
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LS.5
Theory
of
Linear
Systems
1.
General
linear
ODE
systems
and
independent
solutions.
We
have
studied
the
homogeneous
system
of
ODE’s
with
constant
coeﬃcients,
(1)
x
�
=
A
x
,
where
A
is
an
n
×
n
matrix
of
constants
(
n
= 2
,
3).
We
described
how
to
calculate
the
eigenvalues
and
corresponding
eigenvectors
for
the
matrix
A
,
and
how
to
use
them
to
find
n
independent
solutions
to
the
system
(1).
With
this
concrete
experience
solving
low-order
systems
with
constant
coeﬃcients,
what
can
be
said
in
general
when
the
coeﬃcients
are
not
constant,
but
functions
of
the
independent
variable
t
?
We
can
still
write
the
linear
system
in
the
matrix
form
(1),
but
now
the
matrix
entries
will
be
functions
of
t
:
x
=
a
(
t
)
x
+
b
(
t
)
y
±
�
�
±
a
(
t
)
b
(
t
)
� ±
�
x
x
(2)
y
�
=
c
(
t
)
x
+
d
(
t
)
y
,
y
=
c
(
t
)
d
(
t
)
y
,
or
in
more
abridged
notation,
valid
for
n
×
n
linear
homogeneous
systems,
(3)
x
�
=
A
(
t
)
x
.
Note
how
the
matrix
becomes
a
function
of
t
—
we
call
it
a
“matrix-valued
func-
tion”
of
t
,
since
to
each
value
of
t
the
function
rule
assigns
a
matrix:
a
(
t
0
)
b
(
t
0
)
t
0
�
A
(
t
0
) =
c
(
t
0
)
d
(
t
0
)
In
the
rest
of
this
chapter
we
will
often
not
write
the
variable
t
explicitly,
but
it
is
always
understood
that
the
matrix
entries
are
functions
of
t
.
We
will
sometimes
use
n
=
2
or
3
in
the
statements
and
examples
in
order
to
simplify
the
exposition,
but
the
definitions,
results,
and
the
arguments
which
prove
them
are
essentially
the
same
for
higher
values
of
n
.
Definition
5.1
Solutions
x
1
(
t
)
, . . .
,
x
n
(
t
)
to
(3)
are
called
linearly
dependent
if
there
are
constants
c
i
,
not
all
of
which
are
0,
such
that
(4)
c
1
x
1
(
t
)
+
. . .
+
c
n
x
n
(
t
) = 0
,
for
all
t.
If
there
is
no
such
relation,
i.e.,
if
(5)
c
1
x
1
(
t
)
+
. . .
+
c
n
x
n
(
t
)
=
0
for
all
t
�
all
c
i
=
0
,
the
solutions
are
called
linearly
independent
,
or
simply
independent
.
The
phrase
“for
all
t
”
is
often
in
practice
omitted,
as
being
understood.
This
can
lead
to
ambiguity;
to
avoid
it,
we
will
use
the
symbol
�
0
for
identically
0
,
meaning:
“zero
for
all
t
”;
the
symbol
→�
0
means
“not
identically
0”,
i.e.,
there
is
some
t
-value
for
which
it
is
not
zero.
For
example,
(4)
would
be
written
21

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�
�
�
22
18.03
NOTES:
LS.
LINEAR
SYSTEMS
c
1
x
1
(
t
)
+
. . .
+
c
n
x
n
(
t
)
�
0
.

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