Inhomogeneous Linear nth Order
Differential Equations; The Method of
Undetermined Coefficients
We are ready now to discuss
n
th order linear inhomogenous
differential equations. In general these take the form
d
n
y
dx
n
+
p
1
(
x
)
d
n

1
y
dx
n

1
+
...
+
p
n

1
(
x
)
dy
dx
+
p
n
(
x
)
y
=
g
(
x
)
.
To obtain a general solution we obtain the general solution of
the corresponding homogeneous equation with
g
(
x
) = 0 and
we add to that a particular solution, i.e.,
any
solution,
y
p
(
x
) of
the homogeneous equation. While there are
n
th order versions
of the method of variation of parameters these are very com
plex and we will not treat them here. We will concentrate on
the constant coefficient case and the method of undetermined
coefficients.
Differential Operators
We consider now the nth order lin
ear constant coefficient inhomogeneous equation
L
y
=
d
n
y
dx
n
+
p
1
d
n

1
y
dx
n

1
+
...
+
p
n

1
dy
dx
+
p
n
y
=
g
(
x
)
,
wherein the
p
k
are real constants. The characteristic polynomial
associated with the corresponding homogeneous equation is then
p
(
r
)
≡
r
n
+
p
1
r
n

1
+
...
+
p
n

1
r
+
p
n
.
We have noted that it is also possible to write the left hand
side of the differential equation, using the differential operator
notation, as
L
y
=
D
n
y
+
p
1
D
n

1
y
+
...
+
p
n

1
D
y
+
p
n
y.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Because the operator acting on
y
on the right hand side is the
same as the characteristic polynomial
p
(
r
), except that
r
has
been replaced by
D
, we will call that expression
p
(
D
), write the
previous expression as
p
(
D
)
y
and write the original equation as
p
(
D
)
y
=
g.
In fact, for any polynomial, of any degree
m
, in
r
,
q
(
r
) =
r
m
+
q
1
r
m

1
+
· · ·
+
q
m

1
r
+
q
m
,
we will define
q
(
D
) =
D
m
+
q
1
D
m

1
+
...
+
q
m

1
D
+
q
m
I,
where
I
denotes the identity operator such that
I y
=
y
.
Back to Finite Families
We will suppose again that the
function
g
(
x
) appearing as the inhomogeneous term belongs to
a finite family under differentiation. We recall that this means
for some positive integer
m
the
m
th derivative of
g
(
x
) is a linear
combination of
g
(
x
) itself and its derivatives of order
k,
k
=
1
,
2
, . . . m

1:
g
(
m
)
(
x
) =
c
0
g
(
x
) +
c
1
g
0
(
x
) +
· · ·
+
c
m

1
g
(
m

1)
(
x
)
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 ROBINSON
 Equations

Click to edit the document details