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Help understanding homog. eq.

  1. Oct 27, 2007 #1
    Hi. I'm having a problem understanding how to solve non-linear homogeneous equations. For example, (x[tex]^{}2[/tex]+y[tex]^{}2[/tex])dx +xydy = 0 ; x=1, y=1

    I understand that y=xv, v=y/x, and dy=xdv +vdx

    To sub in,
    x[tex]^{}2[/tex] + (v[tex]^{}2[/tex]x[tex]^{}2[/tex])dx +x[tex]^{}2[/tex]v(vdx+xdv) = 0

    Here's where I get lost:
    x[tex]^{}2[/tex][(1+v[tex]^{}2[/tex])dx +v[tex]^{}2[/tex]dx +xvdv] = 0 =>

    x[tex]^{}2[/tex](1+v[tex]^{}2[/tex])dx +v[tex]^{}2[/tex]dx+xvdv = 0 =>

    1+2v[tex]^{}2[/tex]dx +xvdv = 0 =>

    1+2v[tex]^{}2[/tex]dx = -xvdv =>

    [tex]\int-dx/x[/tex] = int(vdv/1+2v^2) =>

    -ln x = 1/4 ln (1+2v[tex]^{}2[/tex]) +C =>

    ln x[tex]^{}4[/tex] + ln(1+2v[tex]^{}2[/tex]) +4c = A

    I think I'm just getting lost in all the algebraic "cleaning up", but I can't figure it out. Can someone pick it apart for me?
  2. jcsd
  3. Oct 28, 2007 #2


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    THis should read:

    Clean this up as follows:
    That is to say:
    or, divided by x^{2}:

    See if you follow this so far!
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