variation of parameters at higher order

Undetermined coefficients and the annihilator method are fast, but they only handle a short menu of forcings — polynomials, exponentials, sines and cosines. What do you do when the forcing is sec x, ln x, or 1/x, or when the coefficients vary so there is no characteristic polynomial at all? Variation of parameters is the general-purpose tool that always produces a particular solution from the homogeneous solutions, no matter the forcing. At higher order it works exactly as at second order, just with more pieces.

The idea: start from a fundamental set y1, ..., yn of the homogeneous equation, whose general solution is c1 y1 + ... + cn yn. Now allow the constants to VARY — replace each ck by a function uk(x) — and seek a particular solution y_p = u1 y1 + ... + un yn. That is one unknown-per-solution, n unknowns, so you need n equations. The construction imposes n-1 convenient conditions (the derivatives of the uk's, weighted by the yk's and their derivatives, cancel at each order up to n-2) plus a final equation that the top derivatives must satisfy L[y_p] = g. Together these give a linear system for u1', ..., un' whose coefficient matrix is exactly the Wronskian matrix. Solving (by Cramer's rule, say) gives each uk' as a ratio of determinants, and integrating recovers the uk's. Assemble y_p.

Variation of parameters matters because it is universal: it needs only a fundamental set and an integrable forcing, so it covers the cases the quick methods cannot, and it works for variable coefficients too. The honest price is real. The formula leaves you with integrals that may be impossible to do in closed form — variation of parameters guarantees a particular solution as an integral, but not necessarily an elementary one — and the bookkeeping at high order, with n equations and a full Wronskian, is heavy. Use the quick methods when they apply; reach for variation of parameters when they do not.