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Thread: 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
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    This is a homogeneous equation because both functions in $\displaystyle dx$ and $\displaystyle dy$ are of degree 2. In this case, make the substitution $\displaystyle y=xv$ and $\displaystyle dy = x dv + v dx$. Then your equation becomes

    $\displaystyle x^2-x^2 v^2 dx + 2 x^3 v dv = 0$

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

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

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