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Math Help - Homogeneous differential equation...

  1. #1
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    Homogeneous differential equation...

    Was wondering if anyone could help me out with this ODE:

    (y-x)dy/dx + (2x+3y) = 0


    I'm told that it's a homogeneous equation, and so presume that the substitution y=ux (i.e. u = y/x) will come into play somewhere. However when I try this it doesn't seem to help too much - is this substitution the right kind of approach or am I missing something?

    Any hints on how to go about this problem would be greatly appreciated.

    Thanks
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  2. #2
    Member kjchauhan's Avatar
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    Let y=ux

    \therefore \frac{dy}{dx} = u + x\frac{du}{dx}

    then ur equation \frac{dy}{dx}=\frac{2x+3y}{x-y} is,

    u + x\frac{du}{dx}=\frac{2+3u}{1-u}

    \therefore x\frac{du}{dx}=\frac{2+3u}{1-u}-u

    \therefore x\frac{du}{dx}=\frac{2+2u+u^2}{1-u}

    \therefore \frac{(1-u)du}{2+2u+u^2}=\frac{dx}{x}

    Now integrate..
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