September 9, 2011
51
5.
Exact Equations, Integrating Factors, and
Homogeneous Equations
Exact Equations
A region
D
in the plane is a connected open set. That
is, a subset which cannot be decomposed into two non
empty disjoint open subsets.
The region
D
is called
simply connected
if it contains
no “holes.” Alternatively, if any two continuous curves in
D
can be continuously deformed into one another. We do
not make this precise here, but rely on standard intuition.
A differential equation of the form
M
(
x, y
)
dx
+
N
(
x, y
)
dy
= 0
(1)
is called
exact
in a region
D
in the plane if the we have
equality of the partial derivatives
M
y
(
x, y
) =
N
x
(
x, y
)
for all (
x, y
)
∈
D
.
If the region
D
is simply connected, then we can find
a function
f
(
x, y
) defined in
D
such that
f
x
=
M,
and
f
y
=
N.
Then, we say that the general solution to (1) is the
equation
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September 9, 2011
52
f
(
x, y
) =
C.
This is because the differential equation can be written
as
df
= 0
.
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 Fall '11
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 Equations, Factors, Sets, general solution, Left Hand Side, Homogeneous Equations Exact Equations

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