between these solutions, we finally find the polar equation of the orbits:
r
=
r
0
e
−
a
(
θ
−
θ
0
)
t/b
.
If you graph this for
a
negationslash
= 0
,
you will get stable or unstable spirals.
Example 2.12.
Consider the specific system
x
′
=
−
y
+
x
y
′
=
x
+
y.
(2.31)
In order to convert this system into polar form, we compute
rr
′
=
xx
′
+
yy
′
=
x
(
−
y
+
x
) +
y
(
x
+
y
) =
r
2
.
r
2
θ
′
=
xy
′
−
yx
′
=
x
(
x
+
y
)
−
y
(
−
y
+
x
) =
r
2
.
This leads to simpler system
r
′
=
r
θ
′
= 1
.
(2.32)
Solving these equations yields
r
(
t
) =
r
0
e
t
, θ
(
t
) =
t
+
θ
0
.
Eliminating
t
from this solution gives the orbits in the phase plane,
r
(
θ
) =
r
0
e
θ
−
θ
0
.
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2.3 Matrix Formulation
43
A more complicated example arises for a nonlinear system of differential
equations. Consider the following example.
Example 2.13.
x
′
=
−
y
+
x
(1
−
x
2
−
y
2
)
y
′
=
x
+
y
(1
−
x
2
−
y
2
)
.
(2.33)
Transforming to polar coordinates, one can show that In order to convert this
system into polar form, we compute
r
′
=
r
(1
−
r
2
)
, θ
′
= 1
.
This uncoupled system can be solved and such nonlinear systems will be
studied in the next chapter.
2.3 Matrix Formulation
We have investigated several linear systems in the plane and in the next
chapter we will use some of these ideas to investigate nonlinear systems. We
need a deeper insight into the solutions of planar systems. So, in this section
we will recast the first order linear systems into matrix form. This will lead
to a better understanding of first order systems and allow for extensions to
higher dimensions and the solution of nonhomogeneous equations later in this
chapter.
We start with the usual homogeneous system in Equation (2.5). Let the
unknowns be represented by the vector
x
(
t
) =
parenleftbigg
x
(
t
)
y
(
t
)
parenrightbigg
.
Then we have that
x
′
=
parenleftbigg
x
′
y
′
parenrightbigg
=
parenleftbigg
ax
+
by
cx
+
dy
parenrightbigg
=
parenleftbigg
ab
cd
parenrightbiggparenleftbigg
x
y
parenrightbigg
≡
A
x
.
Here we have introduced the
coefficient matrix
A
. This is a first order vector
differential equation,
x
′
=
A
x
.
Formerly, we can write the solution as
x
=
x
0
e
At
.
1
1
The
exponential of a matrix
is defined using the Maclaurin series expansion
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 Spring '13
 MRR
 Math, Equations, Constant of integration, Equilibrium point, α

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