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Math Help - Separation of Variables

  1. #1
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    Separation of Variables

    Im having trouble separating functions into the form q(y)dy=p(x)dx. It seems like they should be simple but im stuck on these. Any help would be appreciated. Thanks.

    Here are two examples

    \frac{dy}{dx}=\frac{y^2+1}{xy+y}
    I can only get this far..
    ydy=\frac{y^2+1}{(x+1)}dx

    and im also having trouble with this one.

    \frac{dy}{dx}=xy^2-x-y^2+1
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  2. #2
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    Re: Separation of Variables

    Quote Originally Posted by petenice View Post
    Im having trouble separating functions into the form q(y)dy=p(x)dx. It seems like they should be simple but im stuck on these. Any help would be appreciated. Thanks.

    Here are two examples

    \frac{dy}{dx}=\frac{y^2+1}{xy+y}
    I can only get this far..
    ydy=\frac{y^2+1}{(x+1)}dx

    and im also having trouble with this one.

    \frac{dy}{dx}=xy^2-x-y^2+1
    y\;dy=\frac{y^2+1}{x+1}dx \Rightarrow

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

    for the second one

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

    \frac{dy}{dx}=y^2(x-1)-(x-1)=(y^2-1)(x-1) \Rightarrow

    \frac{dy}{y^2-1}=(x-1)\;dx
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  3. #3
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    Re: Separation of Variables

    Quote Originally Posted by petenice View Post
    I can only get this far..
    ydy=\frac{y^2+1}{(x+1)}dx
    Divide both sides by y^2+1.

    Quote Originally Posted by petenice View Post
    \frac{dy}{dx}=xy^2-x-y^2+1
    xy^2-x-y^2+1 = x(y^2-1)-(y^2-1) = (x-1)(y^2-1)

    Romsek beat me to it.
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  4. #4
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    Re: Separation of Variables

    I see. Yep i figured it was something simple, i just felt i was making up stuff at some point. Too tired to think straight. Thank you so much guys.
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