non-linear differential equation system
I am having trouble with the following question..
dx/dt = y
dy/dt = -x + (1-x^2)y
Find the critical points and determine their stability.
Now, I found the critical point to be (0,0).
The matrix representing the linearized system is
A = [ 0, 1; -1, 0 ] and eigenvalues of this matrix are -i and i, which are complex with the real parts equal to zero, thereofore Linearization theorem can't be used here.
so.. using coordinate shift I get X=x and Y=y
thus dY/dX = -X/Y + (1-X^2).. to solve this I get an integrating factor
e^(0.5*x^2)... which can't be integrated. This is where I get stuck (assuming I have done everything else above correctly)
Does this mean the critical point is unstable and that the orbits of the non-linear system cannot be approximated by the linear approximation?
Or have I done something wrong along the way?
Thanks in advance