# Differential Equations : Phase Lines and Bifurcation Diagrams

Please see the attached file for the fully formatted problems.

22. (a) Use PhaseLines to investigate the bifurcation diagram for the differential equation

....

where a is a parameter. Describe the different types of phase lines that occur.

(b) What are the bifurcation values for the one-parameter family in part (a)?

(c) Use PhaseLines to investigate the bifurcation diagram for the differential equation

....

where r is a positive parameter. How does the bifurcation diagram change from the r = 0 case (see part (a))?

(d) Suppose r is negative in the equation in part (c). How does the bifurcation diagram change?

https://brainmass.com/math/calculus-and-analysis/145380

#### Solution Preview

Attached plots are Figure 1 in 1.jpg and Figure 2 in 2.jpg

(a)

A plot of the RHS ( ay-y^2 ) for a=2 is shown in Figure 1 in BLUE color.

The values y=0 and y = a give equilibrium lines on which y = const.

Since the parabola has a definite negative sign at y^2, the left equilibrium line is always unstable and the right one is always stable, no matter what is the sign of a.

That is trajectories converge to the line y = max(0,a) and diverge ...

#### Solution Summary

Differential Equations, Phase Lines and Bifurcation Diagrams are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.