Fall 2003 Math 308/501–502
6 Theory of HigherOrder Linear ODEs
6.2 Homogeneous Linear Equations
with Constant Coefﬁcients
Fri, 26/Sep
c
±
2003, Art Belmonte
Summary
A homogeneous linear differential equation of order
n
with real
constant coefﬁcients has the form
n
X
k
=
0
a
k
y
(
k
)
(
x
)
=
0. Its
associated
auxiliary
or
characteristic equation
is
n
X
k
=
0
a
k
r
k
=
0.
By the Fundamental Theorem of Algebra, there are
n
roots of this
polynomial equation (accounting for multiplicities), which are real
or occur in complex conjugate pairs. Here is how to ﬁnd solutions
to the differential equation.
•
Let
r
be a real root multiplicity
m
.Then
x
k
e
rx
,
k
=
0
,
1
,...
m

1, are
m
linearly independent
solutions of the DE.
•
Let
r
=
α
+
i
β
be a complex root of the characteristic
polynomial with multiplicity
m
. Then we have the following
2
m
linearly independent solutions of the DE.
x
k
e
α
x
cos
β
x
,
k
=
0
,
1
,...,
m

1
;
x
k
e
α
x
sin
β
x
,
k
=
0
,
1
m

1
.
“Hand” Examples
When necessary, resort to MATLAB for computations.
Example A
Show that
y
1
(
x
)
=
e
x
,
y
2
(
x
)
=
xe
x
,and
y
3
(
x
)
=
x
2
e
x
are
linearly independent.
Solution
Assume these functions are linearly dependent. Then there exist
constants
c
1
,
c
2
,
c
3
, not all zero, such that
c
1
y
1
+
c
2
y
2
+
c
3
y
3
=
0
for all
x
. In particular, for
x
=
1
,
0
,
1, we have
e

1

e

1
e

1
100
eee
c
1
c
2
c
3
=
0
0
0
or
Mc
=
0
, whence
c
=
M
\
0
=
[0
;
0
;
0], a contradiction (since
not all the
c
k
are zero). Hence it must be the case that all the