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October 6th, 2010, 05:01 AM   #1
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Non-Linear First Order ODE: Critical Point Linearization

Question:

dx/dt = x - y + (x^2) - xy

dy/dt = -y + (x^2)

- Determine the critical points for the equation,
- Determine the linearized system for each critical point and discuss whether it can be used
to approximate the behaviour of the non-linear system. What is the type and stability of
each critical point?

Answer:

Hey guys! First post, hope all goes well and there will me many more to come

Ok I am doing this problem and this is the first time I've come across non-linear first order ODE's, usually I have been doing linear ones!

Basically I have established the critical points occur at (0,0) and (1,1) from:

Critical points occur when:

dx/dt = 0 and dy/dt = 0

I am not sure at all how to determine the linearized system for each critical point. I have looked in a couple of text books and online but haven't found too much unfortunately.

Any advice on how to go about this step would be great. If anyone knows of a good place to find an worked example of a similar question that would be great too (I have found this is the best way for me to learn, personally works really well for me!)

Thanks in advance
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October 7th, 2010, 11:40 AM   #2
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Welcome to the forum. If you want help with a problem that has already been posted (especially by you) elsewhere, please just state that you want help and provide a link to the earlier post. In this case, the problem was already posted here, and someone has already posted some help. Was that sufficient or do you want further help here?

Note: many textbooks deal with both linear and non-linear systems, so try using a Google Books search to find explanations and worked examples that will help you.
skipjack is offline  
October 9th, 2010, 07:02 AM   #3
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Re: Non-Linear First Order ODE: Critical Point Linearization

Hey sorry about that. That would have been a much smarter thing to do!

I am still having trouble, particularly finding the type and stability of the critical points away from the origin.

My particular problems are noted here:
http://www.physicsforums.com/showthread.php?p=2921073

Thanks in advance
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