Exponential polynomial solution of a constant-coefficient differential equation ($v(x)={\sum }_j{e}^{{\lambda }_{j}x}{\sum }_{\ell <m_j}{c}_{j\ell }{x}^{\ell }$)

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 ${\sum }_j{m}_j=n$ 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.