How Can Eigenvalues Influence ODE Solutions in Phase Plane Simulations?

[Jhenrique]

The solution's form for the ODE $$\frac{d\vec{r}(t)}{dt\;\;\;\;} = k\;\vec{r}(t)$$ can be generalized like in this diagram: https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg

Exist some program or some way of adjust some program of math for study the behavior of the solution through of the eigenvalues as parameter?

 

[athosanian]

I think mathematica may be helpful, in my memory, it can draw such a picture.