# Making an inexact differential equation exact

#### jumbogala

1. The problem statement, all variables and given/known data
I have another question. The following equation is inexact. Find an integrating factor u that makes the equation exact.

(-9/6)x4y6+3x8y7+x6)dx+(−1x5y5+(13/21)x9y6)dy=0

2. Relevant equations
Call the part before dx M, and the part before dy N.

3. The attempt at a solution
We did an example like this in class where to find u we took [(dM/dy)-(dN/dx)] / N. (those are partial derivatives). Then the answer to that was integrated and put to the power of e (e^integrated answer).

I tried doing that but it gave me something weird.

Partial of M with respect to y: -9x4y5 + 21x8y6

Parital of N with respect to x: -5x4y5 + ((21*9)/13)x8y6

Then take the difference between the two and get -4x4y5 -6.46x8y6

And divide that by N. this gives some weird function that I don't think is right because the answers are usually simple. I'm guessing I wasn't supposed to use the same initial formula that the example did, but all the examples seem to use it.

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#### LCKurtz

Science Advisor
Homework Helper
Gold Member
The test you are trying is to look for an integrating factor a pure function of x. There is a similar test looking for a pure function of y. You can read about these tests in the thread:

https://www.physicsforums.com/showthread.php?p=2375160#post2375160

Neither of these tests seem to work for your problem. Nobody promises that such a DE will have such an integrating factor although many textbook exercises of this type do have them. Sometimes a misprint is to blame when the author intended it to work.

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