When Guessing Fails

Differential Equations · Axiom Academy

Undetermined coefficients is a great trick — until the driving function refuses to play along. Meet the equation that breaks it. The trick behind undetermined coefficients To solve , undetermined coefficients guesses a form for the particular solution y_p built from g(x) and its derivatives. That only works when differentiating g(x) keeps landing on the SAME small handful of function types — a finite family closed under differentiation. Watch two driving functions get differentiated four times each. One keeps cycling through the same two shapes. The other keeps producing a brand-new, more complicated term every single time. repeats after 2 distinct shapes (period 4, sign-alternating). never repeats — each pass adds a strictly higher power of . Which driving functions does the method handle? Click each function below to check: do its derivatives stay inside a small, fixed family — or do they keep spawning new types forever? Every function that "works" for undetermined coefficients shares this one property: differentiation closes. Try every reasonable guess for isn't a polynomial, an exponential, or a sine/cosine — so which trial form could possibly work? Click each candidate guess to test it. A method that never runs out of road Undetermined coefficients is fast when it applies — but it only applies to a short list of driving functions. , , , and plenty of others sit outside that list forever, no matter how clever the guess.

This is the written version of the interactive lesson above. See the full Differential Equations course.