〈 Berger | Dillon 〉

It's worth asking how we know when this kind of factorization can work, given that differential operators don't in general commute...

Exactly! Since the two terms I factored it into commute it doesn't matter.Obviously one can't do this with, say, the Stirm-Liouville operator.. though I think it would be interesting to see if you could generalize something like this to a "quadratic" equation for non commuting Os

Second order DE. I just don’t like operators.

you'll learn to love them

Looks similar to “factoring” (d^2/dt^2- c^2d^2/dx^2) into (d/dt + cd/dx)(d/dt - cd/dx) to get d'Alembert's solution of the wave eq

The Dirac Equation was actually an attempt to take the "square root" of the Klein Gordon equation to get first order time and spatial derivatives