1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    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?

    Last edited: Apr 2, 2012
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted