Why phase diagrams technique can only be used for scalar autonomous Ordinary differential equations and not for non-autonomous ODEs?

Question

Recently, I was trying to understand the concept of Phase diagram. I found that they are only useful in visualizing the solutions of scalar autonomous ODEs and not suitable for investigating non-autonomous ODEs. My questions are,

1. Why we can use the phase diagram technique only for autonomous equations and not for non-autonomous equations?
2. Is there any other method available to visualize the solutions of non-autonomous ODEs?

Please let me know for any other information. Thanks in Advance.

Answer 1

Let $x_1(t)$ and $x_2(t)$ be two solutions of an autonomous system. If $x_1(t_1)=x_2(t_2)$, then $x_2(t)=x_1(t+t_1-t_2)$. This implies that trajectories do not cross. This is no longer true for non autonomous systems, making it impossible to draw a meaningful phase diagram.