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fluidistic
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Homework Statement
Find the stationary solution(s) of the following system of DE and determine its stability:
[itex]x'=x^2+y^2+1[/itex].
[itex]y'=2xy[/itex].
Homework Equations
[itex]x'=\frac{dx}{dt}=0[/itex], [itex]y'=\frac{dy}{dt}=0[/itex].
The Attempt at a Solution
I tried to google "stationary solutions of a system of DE" but didn't find anything that can help me. I'm guessing they mean solutions that does not change with respect to time, hence [itex]x'=y'=0[/itex]. By setting this constraint, I reached that [itex](x,y)=(\pm i \sqrt {1+y^2},0)=(\pm i , 0)[/itex] are critical points. Namely [itex](x,y)=(i,0)[/itex] and [itex](x,y)=(-i,0)[/itex] are 2 critical points, or stationary solutions to the system of DE.
Now I do not know about the stability of such a system. What should I check for?
Thanks in advance!