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Math Help - Homogeneous equations always do have an integrating factor

  1. #1
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    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.
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  2. #2
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    If the ODE is homogeneous then it admits the symmetry

     <br />
\bar{x} = e^{\varepsilon \alpha}x,\;\;\;\bar{y} = e^{\varepsilon \alpha}y,<br />

    or infinitesimal transformation

     <br />
\bar{x} = x + \varspsilon \alpha x + O(\varepsilon^2),\;\;\;<br />
\bar{y} = y + \varspsilon \alpha y + O(\varepsilon^2)<br />

    and the original ODE has the integrating factor

     <br />
\mu = \dfrac{1}{\alpha x M + \alpha y N}<br />
(we can set \alpha = 1 wlog)

    This was show by Lie in the late 1800's.
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  3. #3
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    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.
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  4. #4
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    Quote Originally Posted by Danny View Post
    If the ODE is homogeneous then it admits the symmetry

     <br />
\bar{x} = e^{\varepsilon \alpha}x,\;\;\;\bar{y} = e^{\varepsilon \alpha}y,<br />

    or infinitesimal transformation

     <br />
\bar{x} = x + \varspsilon \alpha x + O(\varepsilon^2),\;\;\;<br />
\bar{y} = y + \varspsilon \alpha y + O(\varepsilon^2)<br />

    and the original ODE has the integrating factor

     <br />
\mu = \dfrac{1}{\alpha x M + \alpha y N}<br />
(we can set \alpha = 1 wlog)

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

    Quote Originally Posted by HallsofIvy View Post
    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.
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