11
Higher-Order Equations: Extending
First-Order Concepts
Let us switch our attention from first-order differential equations to differential equations of
order two or higher. Our main interest will be with second-order differential equations, both
because it is natural to look at second-order equations after studying first-order equations, and
because second-order equations arise in applications much more often than do third-, or fourth-
or eighty-third-order equations. Some examples of second-order differential equations are
1
y
′′
+
y
=
0
,
y
′′
+
2
xy
′
−
5 sin
(
x
)
y
=
30
e
3
x
,
and
(
y
+
1
)
y
′′
=
(
y
′
)
2
.
Still, even higher order differential equations, such as
8
y
′′′
+
4
y
′′
+
3
y
′
−
83
y
=
2
e
4
x
,
x
3
y
(
iv
)
+
6
x
2
y
′′
+
3
xy
′
−
83 sin
(
x
)
y
=
2
e
4
x
,
and
y
(
83
)
+
2
y
3
y
(
53
)
−
x
2
y
′′
=
18
,
can arise in applications, at least on occasion. Fortunately, many of the ideas used in solving
these are straightforward extensions of those used to solve second-order equations. We will make
use of this fact extensively in the following chapters.
Unfortunately, though, the methods we developed to solve first-order differential equations
are of limited direct use in solving higher-order equations. Remember, most of those methods
were based on integrating the differential equation after rearranging it into a form that could
be legitimately integrated. This rarely is possible with higher-order equations, and that makes
solvinghigher-orderequationsmoreofachallenge. Thisdoesnotmeanthatthoseideasdeveloped
in previous chapters are useless in solving higher-order equations, only that their use will tend
to be subtle rather than obvious.
Still, there are higher-order differential equations that, after the application of a simple
substitution, can be treated and solved as first-order equations. While our knowledge of first-
order equations is still fresh, let us consider some of the more important situations in which this is
1
For notational brevity, we will start using the ‘prime’ notation for derivatives a bit more. It is still recommended,
however, that you use the ‘
d
/
dx
’ notation when finding solutions just to help keep track of the variables involved.
241

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242
Higher-Order Equations: Extending First-Order Concepts
possible. We will also take a quick look at how the basic ideas regarding first-order initial-value
problems extend to higher-order initial-value problems. And finally, to cap off this chapter, we
will briefly discuss the higher-order extensions of the existence and uniqueness theorems from
section 3.3.
11.1
Treating Some Second-Order Equations as
First-Order
Suppose we have a second-order differential equation (with
y
being the yet unknown function
and
x
being the variable). With luck, it is possible to convert the given equation to a first-order
differential equation for another function
v
via the substitution
v
=
y
′
. With a little more luck,
that first-order equation can then be solved for
v
using methods discussed in previous chapters,
and
y
can then be obtained from
v

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