A. Alaca
MATH 1005
Winter 2010
8
Integrating factor for nonexact diﬀerential equations
It is sometimes possible to convert a nonexact DE into an exact DE by multiplying
it an integrating factor
I
(
x, y
):
P
(
x, y
)+
Q
(
x, y
)
y
±
=0
(
*
)
(nonexact.)
I
(
x, y
)
P
(
x, y
I
(
x, y
)
Q
(
x, y
)
y
±
= 0
(exact)
∂
∂y
(
IP
)=
∂
∂x
(
IQ
)or
I
y
P
+
y
=
I
x
Q
+
x
or
PI
y

QI
x
=
I
(
Q
x

P
y
)(
**
)
In general, to solve (
**
) is harder than to solve (
*
).
But if
I
=
I
(
x
) (or
I
=
I
(
y
)), then we can solve (
**
).
If
I
=
I
(
x
), then
I
y
= 0 and (
**
) becomes

QI
x
=(
Q
x

P
y
)
I
=
⇒
dI
dx
=
P
y

Q
x
Q
I
=
⇒
dI
I
=
P
y

Q
x
Q
dx.
After we integrate and rise both side to
e
, we obtain

I

=
e
±
1
Q
(
P
y

Q
x
)
dx
.
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