2.1.5 Inexact Equation—Reducible to Exact Equation
by Integrating Factors
Integrating factor (I.F.)
If the differential equation 0MdxNdy+= becomes exact when we
multiply it b
y a function
(,)xy
μ
then
(,)xy
μ
is called an integrating
factor of the equation.
Consider the equation
0yd
xxdy−=
Here
,11
MN
MyNx
yx
∂∂
==−⇒=≠=−
∂∂
The equation is not exact. If we multiply it by
2
1
y
we get
2
1
0.
x
dxdy
yy
−=
Now
2
1
,
x
y
y
MN==−
so that
2
1
.
MN
yyx
∂∂
=−=
∂∂
So, the equation becomes exact.
2
1
y
is an integrating factor. We can eas-
ily check that
222
111
,,
xy
xxy+
are also integrating factors for the equation.
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