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December 17th, 2009, 03:46 AM   #1
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ODE question 2

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 solution ( x(t), y(t) ) of this system, such as: x(t)*y(t) != 1 (doesn't equal...) , there exists a constant 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 derivative 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 derivative 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 derivative of u(t)?

TNX a lot!
WannaBe is offline  
 
December 17th, 2009, 05:01 AM   #2
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For xy ?1, du/dt = 2xx' + 2yy' - (xy' + yx')/(xy - 1)
= 2x(x + y - xy) + 2y(-x - y + xy) - 2(x(-x - y + xy) + y(x + y - xy))/(xy - 1)
= 2x - 2y - 2(-x + xy + y - xy)/(xy - 1),
= 2x - 2y -2(x - y)(xy - 1)/(xy - 1)
= 0.
Hence u(x, y) is a constant.

Other solutions are x = y = 0 and (x, y) = (Ae^t, (1/A)e^(-t)), where A is a non-zero constant.
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December 17th, 2009, 06:30 AM   #3
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Re: ODE question 2

Wow, so we don't even need to solve the the system... Nice one...

Tnx a lot!
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