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Homework Help: Homogeneous Differential Equation

  1. Feb 25, 2010 #1
    (Moderator's note: thread moved from "Differential Equations")

    Hello :)

    I'm trying to solve an homogeneous equation... but it seems that i'm wrong in some step... or something, because I can't complete this problem, look here is what i got:

    [tex]x\frac{dy}{dx} = ye^\frac{x}{y} - x[/tex]

    [tex]x dy = (ye^\frac{x}{y} - x) dx[/tex]

    [tex]x dy + ( -ye^\frac{x}{y} + x) dx[/tex]

    Using substitution
    [tex]x = vy[/tex]

    [tex]dx = vdy + ydv[/tex]

    [tex]vy dy + (-ye^v + vy) [ vdy + ydv ] = 0[/tex]

    [tex](vy + yv^2 - vye^v)dy + (vy^2 - y^2e^v )dv = 0[/tex]

    [tex](vy + yv^2 - vye^v)dy = (y^2e^v - vy^2 )dv[/tex]

    [tex]\frac{(y)(v + v^2 - ve^v)}{y^2(e^v - v)} dy = dv[/tex]

    [tex]\int \frac{1}{y} dy = \int \frac{(e^v - v)}{(v + v^2 - ve^v)} dv[/tex]

    The integral on the left side is easy to solve, but I can't find a way to solve the right side of the equation.

    Any Suggestions?

    Thanks and sorry for my English, I'm still learning it (i'm from venezuela)
    Last edited by a moderator: Feb 26, 2010
  2. jcsd
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