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Integrating Factors for Differential Equations: Example 4
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Solve the differential equation,
, by testing whether an integrating factor can be found. If it can be found, complete the solution.
What is the general form of this differential equation?.
What is a condition for determining an integrating factor?
One condition is
Find
.
Find
Determine the above condition.
Is this a function of only
x
?
No.
What is our conclusion?
This condition does not show that there is an integrating factor.
State the other condition for determining an integrating factor?
The second condition is
Test this condition.
Is this a function of only y?
Yes.
What is our conclusion?
That there is an integrating factor.
How do we find it?
The integrating factor is
Lets focus on the integral
. Set it up.
We get
Evaluate this integral.
Use this result to set up the integrating factor.
We get
Simplify this result.
Since
e
and ln are inverses of each other, we get an
integrating factor
of
What do we do with this integrating factor?
We multiply the original differential equation by it.
Do that.
Our
modified differential equation
becomes
Is this now an exact differential equation?
Yes.
How do we proceed?
We partially integrate the
dx
term.
Set up that integral to get the solution
U(x,y)
.
Do the integral.
Why do we get the
f(y)
term?
Because we did a partial integration with respect to
x
.
How do we determine
f(y)
?
We can determine
from this result and compare it with the coefficient of the
dy
term of the modified differential equation. The two must be equal.
Take the partial derivative.
We get
Compare this with the modified differential equation.
We get
.
What can we conclude about
f(y)
?
We see that
How can we use this result?
We can do a partial integration with respect to
y
.
Do it.
Combine these results to obtain the general solution.
is the solution of our modified differential equation.
Check this result by calculating
,
in agreement with the
dx
term in the modified differential equation.
Check by calculating
in agreement with the
dy
term in the modified differential equation.
The end.
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General Contents
Detailed Contents
Index
, by testing whether an integrating factor can be found. If it can be found, complete the solution.