ordinary differential equations – Find integrating factor

I am trying to solve this equation but I can’t really find an integrating factor:
$$underbrace{ycdot(1+x)}_{P}:dx+underbrace{xcdot(1+y)}_{Q}:dy=0$$
I know I must find $mu equiv mu (x, y)$ so that:
$$frac{partial}{partial y}(mu P)=frac{partial}{partial x}(mu Q)$$
Therefore:
$$Pfrac{partial mu}{partial y}+mu frac{partial P}{partial y}=Qfrac{partial mu}{partial x} + mu frac{partial Q}{partial x}$$
I have trouble working out the partials of $mu$. I have tried doing $muequiv mu (epsilon)$ and $epsilon equiv epsilon (x, y)$, therefore:
$$frac{partial mu}{partial y}=frac{partial mu}{partial epsilon} cdot frac{partial epsilon}{partial y}$$
And the same for the $x$. But I don’t know where to go from here. Could someone please help me?

Ordinary Differential Equations – How Do I Find the Integration Factor of this ODE?

I wanted to know if there is a general method to find the integration factor of a given differential equation $$ Pdx + Qdy = 0 $$ For example, I have the following equation
$$ x ^ 3dx + 2 (x ^ 2-xy ^ 2) dy = 0 $$ that's not exactly.

So far I have tried to find the integration factor for all of the following cases:

  1. If the IF is a function of x, I used the test to see if $$ frac { frac { partial P} { partial y} – frac { partial Q} { partial x}} {Q} $$ is just a function of x.
  2. I checked whether $$ frac { frac { partial P} { partial y} – frac { partial Q} { partial x}} {- P} $$ is a function of only y.
  3. Also whether $$ frac { frac { partial P} { partial y} – frac { partial Q} { partial x}} {Q + P} $$ is a function of x-y
  4. Then, when $$ frac { frac { partial P} { partial y} – frac { partial Q} { partial x}} {_ Px + Qy} $$ is a function of xy
  5. If $$ frac { frac { partial P} { partial y} – frac { partial Q} { partial x}} { frac {yQ + xP} {y ^ 2}} $$ is a function of $$ frac {x} {y} $$
  6. And finally when $$ frac { frac { partial P} { partial y} – frac { partial Q} { partial x}} {2 (xQ-yP)} $$ is a function of $$ x ^ 2 + y ^ 2 $$
    And none of the above cases has proven itself in my example. Now I'm not only stuck in this example, but also find the integration factor of every other differential equation in general. Is there a way to know (and find) what form the integration factor takes without having to try all special cases? Any help would be appreciated, thanks.

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Air Travel – Are there active, regularly scheduled commercial flights that are reserved for one gender or other demographic factor?

I read on https://viewfromthewing.com/united-airlines-men-only-executive-service/ (Spiegel):

From 1953 to 1970, United Executive offered men's-only flights between New York and Chicago, and between Los Angeles and San Francisco.

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Are there active, regularly scheduled commercial flights that are reserved for one gender or another demographic factor?

Differential topology – hinders the existence of a globally defined integration factor

To let $ U $ to be an open subset of $ Bbb {R} ^ n $ and take $ omega $ be a nowhere smoother $ 1 $-Form one $ U $. The Frobenius theorem implies that near every point of $ U $, $ omega $ can be written as $ g , { rm {d}} f $ for suitable locally defined smooth functions $ f $ and $ g $ iff $ omega wedge { rm {d}} omega = 0 $. Here's my question (s): For one $ 1 $-form $ omega $ on $ U $ (never disappear and $ omega wedge { rm {d}} omega = 0 $) What is the obstacle to the existence of global smooth functions? $ f $ and $ g $ on $ U $ With $ omega = g , { rm {d}} f $? Can this be formulated as the disappearance of a homotopic invariant of? $ U $? What is a good non-example in which such a global representation of $ omega $ does not exist?