Derive the Integrating Factor for Homogeneous DE

1. Apr 2, 2012

lkh1986

1. The problem statement, all variables and given/known data
I have this statement:
If $$M(x,y)dx+N(x,y)dy=0$$ is a homogeneous DE, then $$μ(x,y)=\frac{1}{xM+yN}$$ is its integrating factor. The problem is, how do we derive this integrating factor?

2. Relevant equations
For homogeneous DE, we have $$f(kx,ky)=k^n*f(x,y)$$
We also have $$\frac{dy}{dx}=-\frac{M(x,y)}{N(x,y)}=-\frac{M(x,xv)}{N(x,xv)}=-\frac{x^pM(1,v)}{x^pN(1,v)}=-\frac{M(1,v)}{N(1,v)}=F(v)=F(\frac{y}{x})$$

$$\frac{M(x,y)}{N(x,y)}dx+dy=0$$ becomes
$$-F(\frac{y}{x})dx+dy=0$$

3. The attempt at a solution
I try to introduce $$μ$$ into the original DE, then I try to derive the factor, which I know the final answer would be $$μ(x,y)=\frac{1}{xM+yN}$$, but I get very complicated formula, which I cannot simplify. I suspect that there're more properties for homogeneous DE?

Thanks.

Last edited: Apr 2, 2012
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