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.
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!