Solving y'' + 4y = 3sin

Differential Equations · Axiom Academy

A resonance case — the driving force matches the homogeneous solution Find the general solution to . Nice work — you solved a resonance case by recognizing when the standard trial solution fails and fixing it. Recognizing resonance: when the forcing function is already a solution of the homogeneous equation, the usual trial solution reduces to 0 and can never match the forcing term. Modifying the trial: multiply the standard trial by x (or x^2 for a repeated root) to make it linearly independent from y_h . Why it works: the x -terms introduced by the product rule cancel exactly on substitution, leaving only the derivative terms needed to match the forcing function. Physical meaning: resonance makes the amplitude grow linearly with x — the mechanism behind catastrophic oscillation in undamped systems driven at their natural frequency. General pattern: if the forcing term already appears in y_h , multiply the trial by x^m , where m is the smallest positive integer that restores linear independence. This technique is essential for forced-oscillation problems in physics and engineering — especially undamped systems driven at their natural frequency.

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