Method of undetermined coefficients

Summary

The method of undetermined coefficients finds a particular solution of a linear equation with constant coefficients $L y = b(x)$ when $b$ has a special form. The right side must be a sum of terms $x^k e^{\lambda x}$, $x^k e^{\lambda x} \cos(\omega x)$ and $x^k e^{\lambda x} \sin(\omega x)$. For such a term, one tries a function of the same type, with a general polynomial of degree $k$ and unknown coefficients. If $\lambda$ (or $\lambda + i\omega$) is a root of multiplicity $s$ of the characteristic polynomial, one multiplies the trial function by $x^s$. Substitution into the equation gives linear equations for the unknown coefficients.