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Math Help - Nonlinear Autonomous System

  1. #1
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    Question Nonlinear Autonomous System

    Can I get help on the solution of the following 2 systems:
    1. dx/dt = -y(y-1) dy/dt = (x-1)(y-1)
    2. dx/dt = 1/y dy/dt = 1/x
    Thanka a lot for any help.
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  2. #2
    Member kjchauhan's Avatar
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    (1) Pl Note that,

    \frac{dy}{dx}=\frac{dy}{dt} \cdot \frac{dt}{dx}

    \therefore \frac{dy}{dx}=\frac{(x-1)(y-1)}{-y(y-1)}

    \therefore y \cdot dy= -(x-1) \cdot dx

    \therefore y \cdot dy+(x-1) \cdot dx = 0

    Now integrate..

    Same way,,,

    (2)

    \frac{dy}{dx}=\frac{dy}{dt} \cdot \frac{dt}{dx}

    \therefore \frac{dy}{dx}=\frac{y}{x)}

    \therefore \frac{dy}{y}= \frac{dx}{x}

    Now integrate..
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  3. #3
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    Question

    Thank you very much.
    I am able to solve x(t) and y(t) for System #2.
    However, for System #1 if I integrate, I get y^2 = -(x^2)/2+x, and I am not sure how to get y(t) and x(t).
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  4. #4
    Member kjchauhan's Avatar
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    Quote Originally Posted by kjchauhan View Post
    (1) Pl Note that,

    \frac{dy}{dx}=\frac{dy}{dt} \cdot \frac{dt}{dx}

    \therefore \frac{dy}{dx}=\frac{(x-1)(y-1)}{-y(y-1)}

    \therefore y \cdot dy= -(x-1) \cdot dx

    \therefore y \cdot dy+(x-1) \cdot dx = 0

    Now integrate..

    Same way,,,

    (2)

    \frac{dy}{dx}=\frac{dy}{dt} \cdot \frac{dt}{dx}

    \therefore \frac{dy}{dx}=\frac{y}{x)}

    \therefore \frac{dy}{y}= \frac{dx}{x}

    Now integrate..

    y \cdot dy+(x-1) \cdot dx = 0

    \therefore \frac{y^2}{2} + \frac{(x-1)^2}{2} = C(Constant)
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  5. #5
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    Thumbs up

    Thank you very much.
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