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Inhomogeneous Linear Second Order Diferential
Equations
Properties of Solutions;
the General Solution
We turn our
attention now to
linear inhomogeneous second order equations
, which
we may assume written in the form
d
2
y
dx
2
+
p
(
x
)
dy
dx
+
q
(
x
)
y
=
g
(
x
)
,
where
p
(
x
)
, q
(
x
) and
g
(
x
) are known continuous, or piecewise contin
uous, functions deFned on some interval
a
<
x
<
b
.
Since we are
treating the inhomogenous case, the assumption is that
g
(
x
) is not
identically equal to 0 on that interval.
We will begin by listing some
properties of solutions.
•
1)
If
y
1
(
x
) is a solution of the inhomogeneous equation above and
z
(
x
) is a solution of the homogeneous equation
d
2
z
dx
2
+
p
(
x
)
dz
dx
+
q
(
x
)
z
= 0
,
then
y
2
(
x
)
=
y
1
(
x
) +
z
(
x
) is also a solution of the inhomogeneous
equation.
•
2)
If
y
1
(
x
) and
y
2
(
x
) are solutions of the inhomogeneous equation
just indicated, then the di±erence,
z
(
x
) =
y
2
(
x
)

y
1
(
x
), is a solution
of the corresponding homogeneous equation.
Equivalently, there is a
solution
z
(
x
) of the homogeneous equation such that
y
2
(
x
) =
y
1
(
x
) +
z
(
x
).
•
3)
If
g
(
x
)
=
α g
1
(
x
) +
β g
2
(
x
) and
y
1
(
x
) and
y
2
(
x
) are solutions
of the homogeneous equation with
g
(
x
) replaced by
g
1
(
x
) and
g
2
(
x
),
respectively, then
y
(
x
) =
α y
1
(
x
) +
β y
2
(
x
) is a solution of
d
2
y
dx
2
+
p
(
x
)
dy
dx
+
q
(
x
)
y
=
g
(
x
) =
α g
1
(
x
) +
β g
2
(
x
)
.
1
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4)
Suppose the general solution of the homogeneous equation takes
the form
z
(
x, c
1
, c
2
) =
c
1
z
1
(
x
) +
c
2
z
2
(
x
)
,
where
c
1
and
c
2
are arbitrary constants and
z
1
(
x
) and
z
2
(
x
) are two so
lutions of the homogeneous equation. Let
y
p
(
x
) be a particular solution
(i.e.,
any
particular solution) of the inhomogeneous equation. Then the
general solution of the inhomogenous equation is given by
y
(
x, c
1
, c
2
) =
c
1
z
1
(
x
) +
c
2
z
2
(
x
) +
y
p
(
x
)
.
The Frst three of these properties are more or less evident from the
linear form of the equation and require no special proof.
The fourth
property does merit some demonstration.
Proof of 4)
.
Suppose for some
x
0
in the interval
a
<
x
<
b
and
given values
y
0
and
y
1
we are required to satisfy the conditions
y
(
x
0
) =
y
0
,
y
0
(
x
0
) =
y
1
.
Then, using the indicated solution form, we need to choose
c
1
and
c
2
such that
c
1
z
1
(
x
0
) +
c
2
z
2
(
x
0
) +
y
p
(
x
0
) =
y
0
,
c
1
z
1
0
(
x
0
) +
c
2
z
2
0
(
x
0
) +
y
p
0
(
x
0
) =
y
1
.
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 Spring '10
 ROBINSON
 Equations

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