Making an inexact differential equation exact

[jumbogala]

Homework Statement

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

Homework Equations

Call the part before dx M, and the part before dy N.

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.

 

[LCKurtz]

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.