Solving y'' + y = sec

Differential Equations · Axiom Academy

A complete application of variation of parameters — the method undetermined coefficients can't reach. Find the general solution of . Since has no finite family of derivatives (it isn't a polynomial, exponential, or sine/cosine), undetermined coefficients cannot guess a trial form here — variation of parameters is the only method that reaches it. Excellent work! You've applied variation of parameters to a forcing function that undetermined coefficients simply cannot handle. Here's what made it work: Complementary solution first: solve the homogeneous equation y'' + y = 0 to get . The Wronskian is key: for , , the Wronskian W = 1 — about as simple as it gets. The variation-of-parameters formulas: and , where f(x) is the right-hand side. Why this needed variation of parameters: has infinitely many distinct derivatives, so there's no finite trial form to guess — undetermined coefficients is stuck before it starts. Variation of parameters handles ANY continuous forcing function f(x) — polynomial, exponential, trig, or (like here) something with no finite family of derivatives at all.

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