Understanding the Role of Singularities in Finding Integrating Factors

[swimmingtoday]

Suppose you have an equation:

M(x,y) dx + N(x,y) dy = 0

I have heard that there always exists an integrating factor u(x,y) such that the partial derivative of uM with respect to y equals the partial derivative of uN with respect to x.

But somewhere in the back of my mind I remember that there is a condition that the guarantee of the existence of the integrating factor is valid ONLY if there are no singularities in the region.

Can someone please tell me the exact status regarding singularities is? Thank you very much. I appreciate it a lot.

 

[mathwonk]

smart aleck answer: u=0 seems to work.

 

[CPL.Luke]

there isn't always an integrating factor that works, just most of the time.

assuming your not a smart aleck

 

[swimmingtoday]

<<there isn't always an integrating factor that works, just most of the time>>

Thanks.

Is there any particular rule for when there will be one that works? Like for example, is there a rule that an integrating factor exists unless there is a singularity?

I've heard both:

1) An integrating factor ALWAYS exists (but might not neccessarily be easily be found)

and

2) Exists as long as there is no singularity.

 

[CPL.Luke]

I'm going to guess that the 2nd one is true.I'm not sure as in my course we got far enough to know that a single variable integrating factor may not exist.

Although come to think of it we didn't explore multivariable integrating factors, so maybe those always exist, I'll leave it to someone a bit more knowledgeable to answer.

 

[selfAdjoint]

<<there isn't always an integrating factor that works, just most of the time>>

Thanks.

Is there any particular rule for when there will be one that works? Like for example, is there a rule that an integrating factor exists unless there is a singularity?

I've heard both:

1) An integrating factor ALWAYS exists (but might not neccessarily be easily be found)

and

2) Exists as long as there is no singularity.

The rule I learned is that F(x,y)dx + G(x,y)dy = 0 always has an integrating factor as long as F and G are smooth, like there was an existence proof of this, which however I never saw, but that integrating factors in practice was a whole 'nother question and only a few examples exist, only a very much fewer of which are any real use at all.