a. Find the steady states of the differential equation dx dy

= x 2 + k,

where k is a constant, and determine their stability.

b. Sketch all the steady states as functions of k, drawing sinks as solid lines and sources as dotted lines.
Diagrams like this, where the steady states are plotted as functions of a model parameter, are called bifurcation diagrams. They are a major area of research and are very important in a wide range of application areas.
For example, one of the authors of this book uses bifurcation diagrams extensively in the study of cell biology.

c. Find all the places where the number of steady states changes. (These are called bifurcation points.)