= Exponential polynomial solution of a constant-coefficient differential equation
{title2=$v(x)=\sum_j e^{\lambda_jx}\sum_{\ell<m_j}c_{j\ell}x^\ell$}
= Exponential polynomial solutions of a constant-coefficient differential equation
{synonym}
If a degree-$n$ characteristic <polynomial> has distinct <roots of a polynomial> $\lambda_j$ of multiplicities $m_j$, all homogeneous <distribution> solutions are the displayed combinations. There are $\sum_jm_j=n$ independent coefficients. Repeated <characteristic roots of a constant-coefficient differential equation> account for the polynomial factors multiplying exponentials; conjugate pairs give real sine and cosine combinations when real solutions are desired.
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