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ODE: Confused about a Homogeneous Eqn question

  1. May 26, 2009 #1
    1. The problem statement, all variables and given/known data

    So the problem goes like this:

    Code (Text):
    dy/dx = ( x^2 + xy + y^2 ) / x^2
    a) Show that it is a homogeneous equation.
    b) Let v = y/x and express the eqn in x and v
    c) Solve for y
    2. Relevant equations

    (Included)

    3. The attempt at a solution

    Code (Text):
    a) dy/dx = ( x^2 + xy + y^2 ) / x^2 * [(1/xy) / (1/xy)]
             = [(x/y) + 1 + (y/x)] / (x/y)
    Since RHS is expressed only in terms of y/x, therefore it is homogeneous.

    b) v = y/x
       y = vx
       dy/dv = x
       dy/dx = (dy/dv) (dv/dx) = x (dv/dx)
    .'.dy/dx = [(x/y) + 1 + (y/x)] / (x/y) becomes
       x (dv/dx) = [(1/v) + 1 + v] / (1/v)
                 = v^2 + v + 1
       dx/x = dv/(v^2 + v + 1)
     
    The problem is, when I solve by integrating both sides, I get some gibberish arctan (2v/sqrt(3) + 1)term.
    The solution in the book and solved on wolfram alpha is:
    arctan (y/x) - ln |x| = c

    However, this answer suggests that the equation in b) must've been
    dx/x = dv/(v^2 + 1) <- no v term

    Which is clearly not the case...

    What am I doing wrong??
     
  2. jcsd
  3. May 26, 2009 #2
    [tex] \frac {dy}{dx} = x \frac {dv}{dx} + v [/tex]
     
  4. May 26, 2009 #3
    Thank you!
     
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