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Math Help - Tricky homogenous equation

  1. #1
    Newbie staevobr's Avatar
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    Tricky homogenous equation

    xy^2 y' = y^3 + x^2 \sqrt{x^2 - y^2}

    y' = \frac{y}{x} + \frac{x \sqrt{x^2 - y^2}}{y^2}

    I know that everything on the right side of the equation needs to be in terms of y/x, but I can't find a way to make this happen.
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  2. #2
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    \frac{x\sqrt{x^2 - y^2}}{y^2} = \frac{x}{y}\left(\frac{\sqrt{x^2 - y^2}}{y}\right)

     = \frac{1}{\frac{y}{x}}\left(\frac{\sqrt{x^2 - y^2}}{\sqrt{y^2}}\right)

     = \frac{1}{\frac{y}{x}}\left(\sqrt{\frac{x^2 - y^2}{y^2}}\right)

     = \frac{1}{\frac{y}{x}}\left[\sqrt{\left(\frac{x}{y}\right)^2 - 1}\right].


    Now make the substitution v = \frac{y}{x}, that means y = vx and \frac{dy}{dx} = x\,\frac{dv}{dx} + v.
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