# ODE

• Dec 17th 2009, 04:47 AM
WannaBe
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!
• Dec 17th 2009, 06:59 AM
Jester
Quote:

Originally Posted by WannaBe
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 $\frac{du}{dt} = 0$.
• Dec 17th 2009, 07:29 AM
WannaBe
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?