i've got the dynamical system:
Now I have solved the system and found the general solution:
x(t)=Ae(^-2t)v + Be(^-t)u, where v=(1,0), u=(0,1), A,B=constants
Now, it can be shown that dy/dx=y/2x, hence dy/dx=0 only when y=0, and for positive x and y dy/dx>0
however, as x=Ae(^-2t), and y=Be(^-t), we have that if t is negative, x is larger than y, and if t is positive, y is larger than x, implying dy/dx = 0 at some point x>0,y>0 (assuming for example the 2 constants are positive and equal).
How is this possible?