13
Reduction of Order
We shall take a brief break from developing the general theory for linear differential equations
to discuss one method (the “reduction of order method”) for finding the general solution to any
linear differential equation. In some ways, this method may remind you of the material in chapter
11. Indeed, part of the method involves solving a higherorder equation via firstorder methods
as discussed in chapter 11. The general theory developed in chapter 12 will not, however, be
used to any great extent. Instead, the material developed here will help us finish that general
theory (at least partially confirming the suspicions raised at the end of the chapter), and will help
lead us to the complete result on constructing general solutions from particular solutions.
But why worry about completing that general theory if any linear differential equation can
be completely solved by this “reduction of order method”? Because this method requires that
one solution to the differential equation already be known. This limits the method’s applicability.
Also, serious practical difficulties arise when the differential equation to be solved is of order
three or more. Still, there are situations where the method is of practical value, and it will help
us confirm suspicions we already have about general solutions.
Oh yes, there is another reason to develop this method: A rather powerful method for solving
nonhomogeneous equations, the “variation of parameters” method described in chapter 23, is
simply a clever refinement of the reduction of order method.
13.1
The General Idea
The “reduction of order method” is a method for converting any linear differential equation to
another linear differential equation of lower order, and then constructing the general solution
to the original differential equation using the general solution to the lowerorder equation. In
general, to use this method with an
N
th
order linear differential equation
a
0
y
(
N
)
+
a
1
y
(
N
−
1
)
+ · · · +
a
N
−
2
y
′′
+
a
N
−
1
y
′
+
a
N
y
=
g
,
we need one known nontrivial solution
y
1
=
y
1
(
x
)
to the corresponding homogeneous differ
ential equation. We then try a substitution of the form
y
=
y
1
u
where
u
=
u
(
x
)
is a yet unknown function (and
y
1
=
y
1
(
x
)
is the aforementioned known
solution). Plugging this substitution into the differential equation then leads to a linear differential
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284
Reduction of Order
equation for
u
. As we will see, because
y
1
satisfies the homogeneous equation, the differential
equation for
u
ends up being of the form
A
0
u
(
N
)
+
A
1
u
(
N
−
1
)
+ · · · +
A
N
−
2
u
′′
+
A
N
−
1
u
′
=
g
— remarkably, there is no “
A
N
u
” term. This means we can use the substitution
v
=
u
′
,
as discussed in chapter 11, to rewrite the differential equation for
u
as a
(
N
−
1
)
th
order
differential equation for
v
,
A
0
v
(
N
−
1
)
+
A
1
v
(
N
−
2
)
+ · · · +
A
N
−
2
v
′
+
A
N
−
1
v
=
g
.
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 Summer '09
 Differential Equations, Equations, Nonhomogeneous Linear

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