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Math Help - Cant solve this diff equation :X

  1. #1
    Junior Member
    Joined
    Sep 2009
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    46

    Cant solve this diff equation :X

    (x^2-3y^2)dx + 2xydy = 0

    I tried everything I just can't separate the variables :X


    thank you in advance!
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  2. #2
    Junior Member
    Joined
    Sep 2009
    Posts
    29
    This is a homogeneous equation because both functions in dx and dy are of degree 2. In this case, make the substitution y=xv and dy = x dv + v dx. Then your equation becomes

    x^2-x^2 v^2 dx + 2 x^3 v dv = 0

    x^2(1-v^2) dx = -2x^3 v dv

    -\frac{x^2}{2 x^3} dx = \frac{v}{1-v^2} dv

    which is separable. Once you solve for x in terms of v, you can go back and solve the equation by substituting v=\frac{y}{x} and solve for y.
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