I was attempting this problem in Lawrence Perko,
Since the order of eigenvalues do not matter, I found eigenvectors such that the system is diagonalized with 4 then 2.
It is quite easy and my answer matches the one given in the solution booklet.
The problem lies in how the phase portrait is drawn.
See, in both cases, we have different transformation matrices P since the order of the eigenvectors is different. So I agree the official solution is correct and the rotation of the phase portrait makes sense given that particular P.
But my phase portrait looks different now... since its not a simple rotation anymore. I transforms to P when moving from y to x. So the cross and star I marked should move to the circle and starInCircle. I am not clear what went wrong here...
Is my phase portrait wrong? Or does changing the order of eigenvectors affect the axis somehow?
Excellent work on your part and it is great to see you questioning results.
If you do a parametric plot of $(c_1e^{2t},c_2 e^{4t})$, versus $(c_1e^{4t},c_2 e^{2t})$, they will be different.
Hint: are you sure you are drawing the correct phase portrait in the $y-$space?
Hint: The books result is correct.