How to Sketch a Phaseline for Differential Equations?

[SSJVegetto]

Hello all,

I have the following differential equation: $x' = f(x) + a$ with $f(x) = (x+1)^{2}.(1-x)$

Now i have the following questions:
1. Sketch the phaseline for $x' = f(x) + a$ with $a = -2, -1, 0, 1$
Don't calculate the exact intersections but make a qualitative correct picture.

2. Give the values for a where there are 3 equilibrium points. Hint: What are the maxima and
minima of f.

I really am trying but i don't understand how to solve this question and i really need some help on how to do this one.

 

[Eynstone]

For (2), you need to solve x'=0= f(x)+a. Hence, (-a) must lie within the maxima & minima of f, so that the line y=-a cuts the graph of f in 3 distinct points.