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LS.4
Decoupling
Systems
1.
Changing
variables.
A
common
way
of
handling
mathematical
models
of
scientific
or
engineering
problems
is
to
look
for
a
change
of
coordinates
or
a
change
of
variables
which
simplifies
the
problem.
We
handled
some
types
of
firstorder
ODE’s
—
the
Bernouilli
equation
and
the
homogeneous
equation,
for
instance
—
by
making
a
change
of
dependent
variable
which
converted
them
into
equations
we
already
knew
how
to
solve.
Another
example
would
be
the
use
of
polar
or
spherical
coordinates
when
a
problem
has
a
center
of
symmetry.
An
example
from
physics
is
the
description
of
the
acceleration
of
a
particle
moving
in
the
plane:
to
get
insight
into
the
acceleration
vector,
a
new
coordinate
system
is
introduced
whose
basis
vectors
are
t
and
n
(the
unit
tangent
and
normal
to
the
motion),
with
the
result
that
F
=
m
a
becomes
simpler
to
handle.
We
are
going
to
do
something
like
that
here.
Starting
with
a
homogeneous
linear
system
with
constant
coeﬃcients,
we
want
to
make
a
linear
change
of
coordinates
which
simplifies
the
system.
We
will
work
with
n
=
2,
though
what
we
say
will
be
true
for
n
>
2
also.
How
would
a
simple
system
look?
The
simplest
system
is
one
with
a
diagonal
matrix:
written
first
in
matrix
form
and
then
in
equation
form,
it
is
u
�
1
0
u
u
=
�
1
u
(1)
=
,
or
�
.
v
0
�
2
v
v
=
�
2
v
As
you
can
see,
if
the
coeﬃcient
matrix
has
only
diagonal
entries,
the
resulting
“system”
really
consists
of
a
set
of
firstorder
ODE’s,
sidebyside
as
it
were,
each
involving
only
its
own
variable.
Such
a
system
is
said
to
be
decoupled
since
the
variables
do
not
interact
with
each
other;
each
variable
can
be
solved
for
independently,
without
knowing
anything
about
the
others.
Thus,
solving
the
system
on
the
right
of
(1)
gives
�
1
t
�
±
�
±
u
=
c
1
e
1
�
1
t
(2)
,
or
u
=
c
1
e
+
c
2
0
e
�
2
t
.
�
2
t
0
1
v
=
c
2
e
So
we
start
with
a
2
×
2
homogeneous
system
with
constant
coeﬃcients,
(3)
x
�
=
A
x
,
and
we
want
to
introduce
new
dependent
variables
u
and
v
,
related
to
x
and
y
by
a
linear
change
of
coordinates,
i.e.,
one
of
the
form
(we
write
it
three
ways):
u
a
b
x
u
=
ax
+
by
(4)
u
=
D
x
,
=
c
d
y
,
.
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 Fall '09
 vogan
 Math, Linear Algebra, Decoupling Systems

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