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The Nonhomogeneous Equation
±
Linear Operators and Kernels
We have seen that given a secondorder linear homogeneous equation
y
00
+
p
(
x
)
y
0
+
q
(
x
)
y
= 0
we can always deﬁne a linear operator
L
[
y
] =
y
00
+
p
(
x
)
y
0
+
q
(
x
)
y.
In this terminology, the function
y
(
x
)
is a solution if and only if
L
[
y
(
x
)] = 0
. Given a fundamental
set of solutions
y
1
and
y
2
, we saw that the general solution is
y
(
x
) =
c
1
y
1
+
c
2
y
2
This is just a way of writing down the set of all solutions, that is, all functions
y
(
x
)
such that
L
[
y
(
x
)] = 0
.
Definition
Given a linear operator
L
[
y
]
, the
kernel
of
L
is the set of all functions
y
(
x
)
such
that
L
[
y
(
x
)] = 0
.
I
Example
If
y
1
and
y
2
are a fundamental set of solutions to the homogeneous equation above, the kernel of L
is the set of all functions of the form
y
(
x
) =
c
1
y
1
+
c
2
y
2
.
I
Example
If
L
is the linear operator
L
[
y
] =
d
dx
[
y
]
,
ﬁnd the kernel of
L
.
Solution
Here operator is the derivative operator. The kernel is the set of all functions
y
(
x
)
such that
d
dx
[
y
] =
L
[
y
(
x
)] = 0
.
But the functions whose derivative is zero are precisely the constant functions. Thus the
kernel is the set of all functions of the form
y
(
x
) =
C
where
C
is a constant.
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The Nonhomogeneous Equation
Now suppose we wish to solve the nonhomogeneous equation
y
00
+
p
(
x
)
y
0
+
q
(
x
)
y
=
g
(
x
)
.
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 '08
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