Proving the Existence of a Constant for ODE Solutions and u(x,y)

[TheForumLord]

Homework Statement

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.

Homework Equations

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 really see what is the deriative of u, as well as how to solve this ODE...
So, I really need your help in:

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

TNX a lot!

 

[Dick]

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.

 

[TheForumLord]

Yep...Inded...
I've managed to solve it...TNX a lot!