7
The Exact Form and General
Integrating Factors
In the previous chapters, we’ve seen how separable and linear differential equations can be solved
using methods for converting them to forms that can be easily integrated. In this chapter, we will
develop a more general approach to converting a differential equation to a form (the “exact form”)
that can be integrated through a relatively straightforward procedure. We will see just what it
means for a differential equation to be in exact form and how to solve differential equations in
this form. Because it is not always obvious when a given equation is in exact form, a practical
“test for exactness” will also be developed. Finally, we will generalize the notion of integrating
factors to help us find exact forms for a variety of differential equations.
The theory and methods we will develop here are more general than those developed earlier
for separable and linear equations. In fact, the procedures developed here can be used to solve
any separable or linear differential equation (though you’ll probably prefer using the methods
developed earlier). More importantly, the methods developed in this chapter can, in theory at
least, be used to solve a great number of other firstorder differential equations. As we will see
though, practical issues will reduce the applicability of these methods to a somewhat smaller (but
still significant) number of differential equations.
By the way, the theory, the computational procedures, and even the notation that we will
develop for equations in exact form are all very similar to that often developed in the later part of
many calculus courses for twodimensional conservative vector fields. If you’ve seen that theory,
look for the parallels between it and what follows.
7.1
The Chain Rule
The exact form for a differential equation comes from one of the chain rules for differentiating
a composite function of two variables. Because of this, it may be wise to briefly review these
differentiation rules.
First, suppose
φ
is a differentiable function of a single variable
y
(so
φ
=
φ(
y
)
), and that
y
, itself, is a differentiable function of another variable
t
(so
y
=
y
(
t
)
). Then the composite
function
φ(
y
(
t
))
is a differentiable function of
t
whose derivative is given by the (elementary)
chain rule
d
dt
[
φ(
y
(
t
))
]
=
φ
′
(
y
(
t
))
y
′
(
t
)
.
129
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130
The Exact Form and General Integrating Factors
A less precise (but more suggestive) description of this chain rule is
d
dt
[
φ(
y
(
t
))
]
=
d
φ
dy
dy
dt
.
!
◮
Example 7.1:
Let
y
(
t
)
=
t
2
and
φ(
y
)
=
sin
(
y
)
.
Then
φ(
y
(
t
))
=
sin
(
t
2
)
,
and
d
dt
sin
(
t
2
)
=
d
dt
[
φ(
y
(
t
))
]
=
d
φ
dy
dy
dt
=
d
dy
[sin
(
y
)
]
·
d
dt
bracketleftbig
t
2
bracketrightbig
=
cos
(
y
)
·
2
t
=
cos
(
t
2
)
2
t
.
(In practice, of course, you probably do not explicitly write out all the steps listed above.)
Now suppose
φ
is a differentiable function of two variables
x
and
y
(so
φ
=
φ(
x
,
y
)
),
while both
x
and
y
are differentiable functions of a single variable
t
(so
x
=
x
(
t
)
and
y
=
y
(
t
)
). Then the composite function
φ(
x
(
t
),
y
(
t
))
is a differentiable function of
t
, and its
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 Summer '09
 Differential Equations, Equations, Factors, Derivative, µ, exact form

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