the annihilator method
The method of undetermined coefficients asks you to GUESS the right shape of a particular solution, then fix up the guess if it clashes with the homogeneous solutions. That works, but the guessing and the modification rule can feel like folklore. The annihilator method replaces the guesswork with a procedure: it derives the correct trial shape — modification and all — mechanically, using nothing but operator algebra.
The key idea is the annihilator: for a forcing g(x) of the friendly kind (polynomials, exponentials, sines, cosines, and their products), there is a constant-coefficient operator A that kills it, meaning A[g] = 0. For example, D annihilates a constant; D^2 annihilates x; (D - a) annihilates e^(ax); (D^2 + b^2) annihilates cos(bx) and sin(bx). Now take the forced equation L[y] = g and hit BOTH sides with A: you get A L[y] = A[g] = 0, a larger HOMOGENEOUS constant-coefficient equation. Solve it by its characteristic roots; among those solutions sit the homogeneous solutions of the original equation (which you discard) and exactly the extra terms that form the correct trial y_p. The leftover terms, with undetermined coefficients, are your trial solution — already carrying any needed factor of x automatically. Substitute back into L[y] = g to pin the coefficients.
So the annihilator method is undetermined coefficients made systematic: it explains WHY the trial shape is what it is, and it produces the modification rule for free, because a forcing that overlaps a characteristic root simply raises that root's multiplicity in A L, which is exactly what attaches the x factor. The same honest boundary applies: it only handles forcings that some constant-coefficient operator can annihilate — things like tan x or 1/x have no such annihilator, so for those you fall back on variation of parameters.
For y'' - y = e^(2x), the annihilator of e^(2x) is (D - 2). Apply it: (D - 2)(D^2 - 1)[y] = 0, whose roots are 2, 1, -1. The roots 1, -1 give the homogeneous part; the new root 2 gives the trial y_p = A e^(2x). Substituting yields A = 1/3, so y_p = (1/3) e^(2x).
Annihilating the forcing converts the forced equation into a homogeneous one that hands you the trial solution.
The annihilator method is a packaging of undetermined coefficients, not a new power — it solves exactly the same restricted class of forcings, just without the guessing.