METHOD OF UNDETERMINED COEFFICIENTS
KIAM HEONG KWA
We would like to construct the general solution of the nonhomoge
neous equation
(1)
ay
00
+
by
0
+
cy
=
g
(
t
)
,
where
a,b,c
∈
R
with
a
6
= 0 and
g
(
t
) is a smooth function, i.e., a
function that is diﬀerentiable as many times as we want, on an open
interval
I
with the property that only a ﬁnitely many number of linearly
independent terms arise as a result of repeated diﬀerentiation. This
means that there is an
N
∈
N
and
N
constants
c
0
,c
1
,
···
,c
N

1
∈
R
such that
(2)
(
c
0
,c
1
,c
2
,
···
,c
N

1
)
6
= (0
,
0
,
0
,
···
,
0)
and
(3)
c
0
g
(
t
) +
c
1
g
0
(
t
) +
c
2
g
00
(
t
) +
···
+
c
N

1
g
(
N

1)
(
t
) = 0
for all
t
∈
I
. It happens that such a function
g
(
t
) can be expressed as
a ﬁnite sum of the following types of functions:
(4)
P
(
t
)
, P
(
t
)
e
αt
, P
(
t
)
e
αt
cos
βt,
and
P
(
t
)
e
αt
sin
βt.
Here
P
(
t
) denotes a generic polynomial with real coeﬃcients, while
α,β
∈
R
are constants.
To calculate the general solution of (1), we ﬁrst show that it suﬃces
to have computed a particular solution of the same equation. This is
the content of the following theorem.
Theorem 1.
1
Let
Y
(
t
)
be any solution of the nonhomogeneous equa
tion
(5)
y
00
+
p
(
t
)
y
0
+
q
(
t
)
y
=
g
(
t
)
,
Date
: January 22, 2011.
1