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Math Help - Differential Equation

  1. #1
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    Differential Equation

    Hey guys, I am wondering if anyone can help me with this question.

    Solve the given differential equation \frac{dy}{dx} = -\frac{2xy+y^2+2}{x^2+2xy}

    Now the methods of solving I have learned so far are, using an integrating factor, separating the equation and the technique for exact equations using partial derivatives. Thanks!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Oblivionwarrior View Post
    Hey guys, I am wondering if anyone can help me with this question.

    Solve the given differential equation \frac{dy}{dx} = -\frac{2xy+y^2+2}{x^2+2xy}

    Now the methods of solving I have learned so far are, using an integrating factor, separating the equation and the technique for exact equations using partial derivatives. Thanks!
    Note that \frac{\,dy}{\,dx}=-\frac{2xy+y^2+2}{x^2+2xy}\implies (x^2+2xy)\,dy=-(2xy+y^2+2)\,dx \implies (2xy+y^2+2)\,dx+(x^2+2xy)\,dy=0

    Can you try to take it from here? It should somewhat be obvious now...

    --Chris
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    Note that \frac{\,dy}{\,dx}=-\frac{2xy+y^2+2}{x^2+2xy}\implies (x^2+2xy)\,dy=-(2xy+y^2+2)\,dx \implies (2xy+y^2+2)\,dx+(x^2+2xy)\,dy=0

    Can you try to take it from here? It should somewhat be obvious now...

    --Chris
    Yea I got to that point, but the equation isn't exact since the partial derivatives are not equal. So I need to find an integrating factor. (Unless I did the partial derivatives wrong and it is exact).
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Oblivionwarrior View Post
    Yea I got to that point, but the equation isn't exact since the partial derivatives are not equal. So I need to find an integrating factor. (Unless I did the partial derivatives wrong and it is exact).
    It is exact:

    M(x,y)=2xy+y^2+2 and N(x,y)=x^2+2xy

    Its exact when \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}

    \frac{\partial M}{\partial y}=\color{red}2x+2y

    \frac{\partial N}{\partial x}=\color{red}2x+2y

    Can you continue now?

    --Chris
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  5. #5
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    Wow it is exact, no wonder. I thought M(x,y) =  x^2 +2xy
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