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Math Help - Solving ODE by inspection

  1. #1
    Junior Member BayernMunich's Avatar
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    Solving ODE by inspection

    Hello

    I stopped at this one:

    y(x^2-y^2+1)dx-x(x^2-y^2-1)dy=0

    I opened the brackets:

    yx^2dx - y^3dx + ydx-x^3dy+xy^2dy+xdy=0

    It will be:

    yx^2dx - y^3dx-x^3dy+xy^2dy+d(xy)=0

    then ?!!!!
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  2. #2
    Junior Member BayernMunich's Avatar
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    I multiplied the last equation by 3, to get:

    yd(x^3)-3y^3dx-3x^3dy+xd(y^3)+3d(xy)=0

    Now, I stopped.
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  3. #3
    MHF Contributor
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    Re-writing gives

    -(x^2-y^2)(xdy-ydx) + xdy+ydx = 0

    -(x^2-y^2)x^2 d\left(\dfrac{y}{x}\right) + d(xy) = 0

    (1-\frac{x^2}{y^2}) x^2y^2d\left(\dfrac{y}{x}\right) + d(xy) = 0

    \left(1-\dfrac{1}{\left(\frac{y}{x}\right)^2}\right) d\left(\dfrac{y}{x}\right) + \dfrac{d(xy)}{(xy)^2} = 0
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  4. #4
    Junior Member BayernMunich's Avatar
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    Thanks ..

    so the final answer will be :

    \dfrac{y}{x}+\dfrac{x}{y}-\dfrac{1}{xy}=0

    Right?
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  5. #5
    MHF Contributor
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    + c but that's what I got.
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