Sections 2.12.2
Integrating factor
dy
dx
+
p
(
x
)
y
=
g
(
x
)
integrating factor
μ
(
x
)
Step 1: Your original equation may not be in this form. Recognize
p
(
x
) and
q
(
x
) after transformation. No sign mistakes!!
Step 2:
Set
μ
(
x
) = exp
p
(
x
)
dx
.
This is an indefinite integral, but we
usually choose the arbitrary constant
c
to be 0 at this step for convenience.
Step 3: New equation
μ
(
x
)
y
=
μ
(
x
)
g
(
x
)
dx
+
c
. Solve for
y
.
Step 4: For initial value problem, if you can not find an explicit solution for
y
, then the best thing to do is to set the lower limit of integration to be the
x
coordinate of the given point, as in Example 4.
Remark
: You will constantly be asked to determine the interval of existence
for your solution. Example 3, a better way to see this is by looking at the given
point (1
,
2), the
x
coodinate 1 is to the positive
y
axis.
Separation
separable form
M
(
x
)
dx
+
N
(
y
)
dy
= 0
separation
Step 1: Your original equation may not be in this form.
Recognize
M
(
x
)
and
N
(
y
) after transformation. No sign mistakes!!
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 Spring '08
 Fonken
 original equation, dy dx

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