Fall 2003 Math 308/501–502
9 Matrix Methods for Linear Systems
9.7M Nonhomogeneous Lin Sys:
Method of Undetermined Coefﬁcients
Mon, 24/Nov
c
±
2003, Art Belmonte
Summary
Method of Undetermined Coefﬁcients for Systems
Let
A
be an
n
×
n
real
constant
matrix and
f
an
n
×
1 column
vector whose elements are sums of products of real polynomials,
sines, cosines, and/or exponentials involving the independent
variable
t
. We may use the method of undetermined coefﬁcients
procedure from Section 6.3 as a
guide
to ﬁnding a particular
solution
x
p
to the nonhomogeneous linear system
x
0
=
Ax
+
f
or
L
[
x
]
=
x
0

Ax
=
f
. The undetermined coefﬁcients involved are
now symbolic
vector
constants.
Moreover, in case an element of
f
is replicated in a general
solution of the associated homogeneous linear system
L
[
x
]
=
0
,
the original choice for a particular solution must
not only
be
multiplied by the smallest positive integer power of
t
so that no
term of the particular solution
x
p
is a solution of the homogeneous
equation
L
[
x
]
=
0
,
but also
by all lower nonnegative integer
powers of
t
as well. This is easier said than done. Indeed, when
this level of complexity is reached, it is simpler to resort to
variation of parameters
, the other technique for ﬁnding
particular solutions that we encountered. This will be discussed
later in lecture handout
9.7V
.
Superposition Principle
For
k
=
1
,
2
,...,
M
,let
x
p
k
be a solution of
L
[
x
]
=
f
k
. Then for
any constants
c
1
c
M
, the function
x
p
=
M
X
k
=
1
c
k
x
p
k
solves the
nonhomogeneous linear system
L
[
y
]
=
M
X
k
=
1
c
k
f
k
. (This follows
immediately from the fact that
L
is a linear operator.)
Hand Examples
In our ﬁrst example, we’ll do things souptonuts by hand. In the
next example, we’ll assume we have the necessary eigenpairs (i.e.,
pairs of eigenvalues with associated eigenvectors) so as to rapidly
form a general solution of the associated homogeneous system.
Then we’ll proceed to the main course: ﬁnding a particular
solution of the nonhomogeneous system.
555/2
Find a general solution to the nonhomogeneous system
x
0
=
Ax
+
f
,where
A
=
±
11
41
²
and
f
=
±

t

1

4
t

2
²
.
Solution
Here is our overall solution strategy.
1. Find a general solution
x
h
to the associated homogeneous
system
x
0
=
Ax
.
2. Find a particular solution
x
p
to the nonhomogeneous system
x
0
=
Ax
+
f
.
3. Form a general solution of the nonhomogeneous system:
x
=
x
p
+
x
h
.
Along the way, we’ll ﬂesh out details. Let’s get the party started.
1.
Computation of x
h
.
(a)
Eigenvalues of A.
Solve det
(
A

r
I
)
=
0.
³
³
³
³
1

r
1

r
³
³
³
³
=
r
2

2
r

3
=
(
r
+
1
)(
r

3
)
=
0,
whence
r
=
1
,
3.
(b)
Associated Eigenvectors.
Find a nonzero vector in the
nullspace of the RREF of
A

r
I
.