General Contents
Detailed Contents
Index
Integrating Factors for Differential Equations: Example 2
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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.
General Contents
Detailed Contents
Index
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?
Yes.
What is our conclusion?
That there is an integrating factor.
How do we find it?
The integrating factor is
Let’s focus on the integral of
Set it up.
We get
Evaluate this integral.
Use this result to set up the integrating factor.
We get
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).
The second term is of the form
. What should we choose for u?
u =
3
x
, or
x = u
/3.
What do we get for
du
?
du=
3
dx,
or
dx
=du/
3
Use these results to rewrite the integral in terms of u in the second term.
Do the integration.
Rewrite this with
u
replaced by its equivalent, 3
x
.
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 “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.
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, by testing whether an integrating factor can be found. If it can be found, complete the solution.