Undetermined Coefficients

A precise trial-function method for particular solutions of constant-coefficient linear equations.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

Undetermined coefficients finds a particular solution of a linear constant-coefficient ODE when the forcing is a finite combination of polynomials, exponentials, sines, cosines, and their products. Choose a trial family closed under differentiation, substitute, and solve algebraically for its coefficients.

Notation and mathematical language

For forcing $e^{\alpha t}P_m(t)\cos\beta t$ or sine, include both sine and cosine with general degree-$m$ polynomials. If the trial overlaps a homogeneous mode whose characteristic root $\alpha+i\beta$ has multiplicity $s$, multiply the entire trial by $t^s$.

Conceptual picture

The operator maps the chosen finite-dimensional trial space into itself. Coefficient matching in that space replaces integration. Resonance means the operator annihilates part of the naive trial, so multiplication by $t$ moves to an independent space.

Fully worked example

Interpretation and application

The method computes forced responses in circuits and vibrations. Resonant algebra reflects a physical or dynamical mode match, but real amplitude conclusions must incorporate damping, parameter uncertainty, and the forcing's duration.