Where a coupling stops being small

Assumed background: separating the variables in a first-order differential equation. How strongly two particles interact is set by a number called a coupling, written $g$. The surprising part is that $g$ is not a constant: it depends on the energy of the particles doing the interacting. An electron's charge measured in a low-energy experiment and the same charge measured at a collider are different numbers. The rate at which a coupling changes with energy is itself calculable, and the calculation produces a differential equation for $g$ with a coefficient fixed by which particles exist in the theory. Following a coupling from one energy to another with that equation is called renormalization group flowThe rate at which a coupling changes with energy scale, $\beta(g) = \mu\, dg/d\mu$, which decides a marginal coupling's fate., and it underlies most of how particle physics relates experiments done at different energies. It is also what says whether a theory written down at low energy can be trusted at high energy, which is the subject of this problem. The equation below keeps only the first term of a series in $g^2$, with terms like $g^5$ on the right-hand side dropped. That approximation is good while $g$ is small. Part (c) explores what happens when $g$ is no longer small.