Characteristic root of a constant-coefficient differential equation
= Characteristic root of a constant-coefficient differential equation
= Characteristic root
{synonym}
For the homogeneous <linear differential equation> $P(\partial_x)u=0$, a characteristic root is a <root of a polynomial> $P(r)=0$. A simple root supplies an exponential solution $e^{rx}$; a root of multiplicity $m$ supplies $x^je^{rx}$ for $0\leq j<m$. The sign of the <real part> of $r$ determines spatial growth or decay.