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Derive the Integrating Factor for Homogeneous DE

  1. Apr 2, 2012 #1
    1. The problem statement, all variables and given/known data
    I have this statement:
    If [tex]M(x,y)dx+N(x,y)dy=0[/tex] is a homogeneous DE, then [tex]μ(x,y)=\frac{1}{xM+yN}[/tex] is its integrating factor. The problem is, how do we derive this integrating factor?



    2. Relevant equations
    For homogeneous DE, we have [tex]f(kx,ky)=k^n*f(x,y)[/tex]
    We also have [tex]\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})[/tex]

    [tex]\frac{M(x,y)}{N(x,y)}dx+dy=0[/tex] becomes
    [tex]-F(\frac{y}{x})dx+dy=0[/tex]

    3. The attempt at a solution
    I try to introduce [tex]μ[/tex] into the original DE, then I try to derive the factor, which I know the final answer would be [tex]μ(x,y)=\frac{1}{xM+yN}[/tex], 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
  2. jcsd
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