I came by a question here regarding the sketching of phase portrait of a two-equations system, and they were referred to a certain tutorial from Paul Online Math Notes. The tutorial ordered to use the representing matrix by taking a certain point (x,y) and find their product (in Av direction of course to get a vector) and the point we get is where some vector in (x,y) is headed. (Not necessary to use a matrix on second thought.)
The point is: it works really well in the tutorial, but using it for $$\dot x=3x,\qquad \dot y =y,$$ I get that, for instance, $(1,y)$ points in the direction of $(3,y)$ which will point towards $(9,y)$ to $(3^n,y)$ no matter what $y$ and using it entirely ignoring my intuition and WolframAlpha graph plotter, I always end up drawing straight lines which is clearly inconsistent with what the graph formally look like in some areas.
I would really appreciate it if you could tell me what I am possibly doing wrong.
"I always end up drawing straight lines": Well, this is odd, see plot below.