polynomial-in-D algebra

Here is a small piece of mathematical sleight-of-hand that pays off enormously. Once you write the derivative as the symbol D, an expression like D^2 - 3D + 2 looks exactly like the ordinary polynomial r^2 - 3r + 2. The bold idea is to TREAT it like one: add, multiply, and factor these D-expressions using the very same rules of algebra you learned in school — and, for constant coefficients, it all works.

Concretely, with constant-coefficient operators you may add term by term, multiply by composing (which mirrors multiplying polynomials), and factor exactly as you would factor a polynomial in r. So D^2 - 3D + 2 factors as (D - 1)(D - 2), meaning 'apply D - 2, then apply D - 1.' The shifting rule makes this calculus easy: D[e^(ax)] = a e^(ax), so any polynomial p(D) acting on e^(ax) just gives p(a) e^(ax) — you literally evaluate the polynomial at a. There is also the exponential-shift identity p(D)[e^(ax) u] = e^(ax) p(D + a)[u], which lets you slide an exponential out of the way and shift the operator's argument. These few moves turn many derivative manipulations into pure algebra.

This is the engine behind operator factorization and the annihilator method, and it is why constant-coefficient theory feels so mechanical and reliable. But hold the algebra honestly: the polynomial-like rules — especially the freedom to reorder and factor — hold ONLY for constant coefficients. With variable coefficients D does not commute with multiplication by x, so 'factoring' is order-sensitive and the clean polynomial analogy breaks. Use the algebra where it is licensed, and switch methods where it is not.