Thread: Homogeneous equations always do have an integrating factor

1. Homogeneous equations always do have an integrating factor

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

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

find an integrating factor for the above equation.

2. If the ODE is homogeneous then it admits the symmetry

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

or infinitesimal transformation

$
\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

$
\mu = \dfrac{1}{\alpha x M + \alpha y N}
$
(we can set $\alpha = 1$ wlog)

This was show by Lie in the late 1800's.

3. 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.

4. Originally Posted by Danny
If the ODE is homogeneous then it admits the symmetry

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

or infinitesimal transformation

$
\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

$
\mu = \dfrac{1}{\alpha x M + \alpha y N}
$
(we can set $\alpha = 1$ wlog)

This was show by Lie in the late 1800's.
two advanced for me! but thanks!

Originally Posted by HallsofIvy
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.
i never said the above ODE was linear or something, it could have quadratic or cubic terms and can still be homogeneous.