I'm looking for some proofs about the following statement:

let $\displaystyle M(x,y)\,dx+N(x,y)\,dy=0,$ be a homogeneous ODE.

find an integrating factor for the above equation.

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- Oct 1st 2010, 08:31 PMKrizalidHomogeneous equations always do have an integrating factor
I'm looking for some proofs about the following statement:

let $\displaystyle M(x,y)\,dx+N(x,y)\,dy=0,$ be a homogeneous ODE.

find an integrating factor for the above equation. - Oct 2nd 2010, 05:29 AMJester
If the ODE is homogeneous then it admits the symmetry

$\displaystyle

\bar{x} = e^{\varepsilon \alpha}x,\;\;\;\bar{y} = e^{\varepsilon \alpha}y,

$

or infinitesimal transformation

$\displaystyle

\bar{x} = x + \varspsilon \alpha x + O(\varepsilon^2),\;\;\;

\bar{y} = y + \varspsilon \alpha y + O(\varepsilon^2)

$

and the original ODE has the integrating factor

$\displaystyle

\mu = \dfrac{1}{\alpha x M + \alpha y N}

$ (we can set $\displaystyle \alpha = 1$ wlog)

This was show by Lie in the late 1800's. - Oct 3rd 2010, 04:38 AMHallsofIvy
Let me point out that "find an integrating factor" is NOT a statement and you are not trying to prove it.

In fact,**any**first order differential equation has an integrating factor- its just simpler to find if the equation is homogeneous. - Oct 3rd 2010, 07:11 AMKrizalid