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Math Help - How to show an equation is homogeneous.

  1. #1
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    How to show an equation is homogeneous.

    If the right side of the equation dy/dx = f(x,y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous.

    (1) \frac{dy}{dx} = \frac{x^2+xy+y^2}{x^2}

    \frac{dy}{dx} = 1 + \frac{y}{x} + \frac{y^2}{x^2}

    That one was easy.

    (2) \frac{dy}{dx} = \frac{4y-3x}{2x-y}

    How do I show that this is homogeneous? I know there is a simple way, but I can't remember for the life of me. Thanks!
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  2. #2
    Super Member Showcase_22's Avatar
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    I had a crack at it and got this far:



    There's probably a better way of doing it than this (mainly because this didn't get an answer).

    hope this helps......somehow.
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  3. #3
    Senior Member bkarpuz's Avatar
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    Solution for (2)

    Quote Originally Posted by shadow_2145 View Post
    If the right side of the equation dy/dx = f(x,y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous.

    (1) \frac{dy}{dx} = \frac{x^2+xy+y^2}{x^2}

    \frac{dy}{dx} = 1 + \frac{y}{x} + \frac{y^2}{x^2}

    That one was easy.

    (2) \frac{dy}{dx} = \frac{4y-3x}{2x-y}

    How do I show that this is homogeneous? I know there is a simple way, but I can't remember for the life of me. Thanks!
    Please see the followings
    \frac{dy}{dx} = \frac{4y-3x}{2x-y}
    ..... =\frac{4y}{2x-y}-\frac{3x}{2x-y}
    ..... =\frac{4}{2\frac{x}{y}-1}-\frac{3}{2-\frac{y}{x}}
    ..... =\frac{4}{2\frac{1}{\frac{y}{x}}-1}-\frac{3}{2-\frac{y}{x}}
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  4. #4
    Math Engineering Student
    Krizalid's Avatar
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    For the second one, just divide by x, top & bottom.
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