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As far as I can see, such an equation typically isn't separable even if p(t) is a constant. For instance, dx/dt + x = t isn't separable. I guess it would be separable if q(t) = c p(t) for some constant c (maybe that is necessary?)

I don't think this step is really necessary to the problem. If the goal is to plot the bifurcation curve, we can just do it directly with Sage implicit_plot (or Desmos, etc) without needing to solve for one variable, nor reflect the curve as in the next step.

In fact, when I assigned this project, most students used implicit_plot to plot the reflected version anyway. But then they tended to think that the reflection was doing something mathematically very important, rather than just working around the restriction of only wanting to plot explicit functions, and so made an unnecessarily big deal about it.

Sources and sinks need a more in depth discussion for repeated eigenvalues.

Should this be + 3 instead of - 3? As it stands, I don't think the equation is separable.

Also appears at

I think this passage could use some more explanation. We describe the equilibrium at 0 as a "nodal source" (or sink), but what do we mean by that? In what ways is it like a node, and in what ways is it like a source? As it's still the case that all solutions diverge from the origin, why is it necessary to distinguish this case from an ordinary source?

Basically, this chapter discusses the algebraic issues around solving the system in the case of a repeated eigenvalue, and how the solution takes a special algebraic form. But I think it would help to have some discussion of how the behavior of such a system differs qualitatively from other cases.