This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and **homogeneous functions of the same degree $d$ and such that $M(x,y)dx+N(x,y)dy$ is not exact** that is:
$$ \frac{\partial M}{\partial y}\neq\frac{\partial N}{\partial x} $$. Using Euler's identity on $M$ and $N$: 


$$ xM\_{x}(x,y)+yM\_{y}(x,y)=d\cdot M(x,y)$$


$$ xN\_{x}(x,y)+yN\_{y}(x,y)=d\cdot N(x,y)$$


**Euler's identity comes from Euler's homogeneous function theorem which is appicable in this case since $M$ and $N$ are both homogeneous functions.**


I showed that the function


$$\mu(x,y)=\frac{1}{xM(x,y)+yN(x,y)}\tag1$$


will satisfy:


$$ \frac{\partial}{\partial y}\left(\,\mu\cdot M\right)=\frac{\partial}{\partial x}\left(\mu\cdot N\right).\tag2$$


My question: If we suppose that the PDE above is true,then how we can explain where the formula of $\mu(x,y)$ comes from?, that is:


$${If}\quad N(x,y)\mu\_{x}-M(x,y)\mu\_{y}=(N\_{x}-M\_{y})\mu,\quad \text{then where the formula} \quad\mu(x,y)=\frac{1}{xM(x,y)+yN(x,y)} \quad \text{comes from ?} $$


I tried to apply the method of characteristics but I don't see it. This result comes from an old edition of Boyce and Di prima.



General formula for integrating factor of an homogeneous differential 1 form
The last display asks whether (1) is the *unique* solution of (2). It isn’t: try $M=N=x$, $\mu=1/x$.


In fact an integrating factor is never unique: see e.g. Serret ([1886, thm 681](https://archive.org/stream/coursdecalculdif02serruoft#page/449)).


Now if you are asking for *heuristics*, then e.g. (ibid., §685) “derives” (1) under the Ansatz that $\mu$ is itself homogeneous (of degree $−d−1$).


