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Math Help - ODE

  1. #1
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    ODE

    Given this ODE:

    x' = x+y-xy^2
    y' = -x-y+x^2y

    and a function: u(x,y) = x^2+y^2-2ln|xy-1|

    prove that for each soloution ( x(t), y(t) ) of this system, such as: x(t)*y(t) != 1 (doesn't equal...) , there exists a constsnt C such as: u ( x(t), y(t) ) = C for every t in R.

    My attempt:
    It's very clear that we need to look at the deriative of u... If it will be 0, then we'll get what we need...But since I haven't got that much knowledge in 2 variables functions, I can't realy see what is the deriative of u, as well as how to solve this ODE...
    So, I realy need your help in:

    1. Solving the ODE.
    2. What is the deriative of u(t)?

    TNX a lot!
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    Given this ODE:

    x' = x+y-xy^2
    y' = -x-y+x^2y

    and a function: u(x,y) = x^2+y^2-2ln|xy-1|

    prove that for each soloution ( x(t), y(t) ) of this system, such as: x(t)*y(t) != 1 (doesn't equal...) , there exists a constsnt C such as: u ( x(t), y(t) ) = C for every t in R.

    My attempt:
    It's very clear that we need to look at the deriative of u... If it will be 0, then we'll get what we need...But since I haven't got that much knowledge in 2 variables functions, I can't realy see what is the deriative of u, as well as how to solve this ODE...
    So, I realy need your help in:

    1. Solving the ODE.
    2. What is the deriative of u(t)?

    TNX a lot!
    \frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt} substitute the derivatives and show \frac{du}{dt} = 0.
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  3. #3
    Member
    Joined
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    Got it... So we don't even need to solve this system...Nice one...

    TNX a lot!

    Can you help me in the other ODE question I've posted?
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