Method of Undetermined Coefficients

GRE Math Subject · Axiom Academy

LESSON Method of Undetermined Coefficients Finding particular solutions to non-homogeneous constant-coefficient ODEs We now solve equations of the form: The general solution has two parts: = complementary solution (general solution of the homogeneous equation ) = particular solution (any single solution of the non-homogeneous equation) You already know how to find from the previous lesson. The method of undetermined coefficients gives you . The method works when is a polynomial, exponential, sine, cosine, or any product/sum of these. The trial solution mirrors the form of : Critical rule: Even if with no cosine, you must include both and in the trial. Example: Polynomial Right-Hand Side Step A -- Find : Characteristic equation: Step B -- Trial solution: Since is a degree-2 polynomial, try: Step C -- Substitute and match: Example: Exponential Right-Hand Side Problem: The natural trial is already part of ! This means substituting it will give 0 on the left side, not . The Modification Rule: If your trial solution (or any term in it) is already a solution of the homogeneous equation, multiply the entire trial by . If it is still a solution, multiply by , and so on. Example: Sine/Cosine Right-Hand Side Problem again: The natural trial duplicates . After substituting and simplifying (which involves product rule twice): Physical note: The term grows without bound -- this is resonance , which occurs when the forcing frequency matches the natural frequency.

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