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ODE question 2

  1. Dec 17, 2009 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    3. The attempt at a solution
    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!
  2. jcsd
  3. Dec 17, 2009 #2


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    Science Advisor
    Homework Helper

    You don't have to solve the ODE. You just have to find d/dt of u(x,y). Then substitute your expressions for dx/dt and dy/dt in and see if you get 0.
  4. Dec 17, 2009 #3
    I've managed to solve it...TNX a lot!
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