The eigen values are 1 The answer key shows but how do you g

The eigen values are -1. The answer key shows

but how do you get that level of detail (or close) by hand. Based on the eigenvalues I can determin the general shape with a given chart but how do you know at what points the curvature of the line changes?

Thanks!

Solution

It is certainly not possible to replicate the accuracy of computer generated images.

Phase portraits are drawn , given the differential equations,; not the solutions.

In fact, phase portraits help understanding the qualitative nature of the solutions, where it may not be possible to obtain explicit solutions (as in your case)

Here are two links that would clarify the picture:

http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx

https://www.youtube.com/watch?v=dpbRUQ-5YWc