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differential equations critical points

A frustratingly simple q. See here:

http://www.maths.ox.ac.uk/system/files/coursematerial/2013/2861/1/DEprobs1a.pdf

Q1 first part.

I understand that as the axes and y=x are trajectories, they can only cross at critical points. However for a trajectory starting inside the open octant, what stopping it from going into one of the critical points on theboundary (like the origin), thereby leaving the open region? Obvious thing would be to find the nature of the critical points and show this is not possible, but we are expected to deduce this before analysing any critical points.

Thanks

Reply 1

ghostwalker

Original post by quint101

Obvious thing would be to find the nature of the critical points and show this is not possible, but we are expected to deduce this before analysing any critical points.

Don't know the answer to your question, but we can note that for small x,y, dx/dt,dy/dt > 0

E.g. when 0 < x,y < 1/3, so it's not going to go through the origin.

If nothing else, this bumps it for someone more knowledgeable to look at.

Reply 2

james22

Linearise about the origin to show that the trajectories go away from it, it doesn't take long. You could also classify it but this takes longer (although you need to later anyway so may be worth doing now).