= Characteristic equation of a constant-coefficient differential equation
{title2=$P(r)=0$}
= Characteristic equation
{synonym}
For the homogeneous <linear ordinary differential equation> $\sum_{j=0}^n a_j y^{(j)}=0$, substitute $y=e^{rx}$. Dividing by the nonzero exponential gives the polynomial equation $P(r)=\sum_j a_jr^j=0$. A simple root gives an exponential solution; a root of multiplicity $m$ gives $x^ke^{rx}$ for $0\le k<m$. Real equations admit real sine and cosine combinations of conjugate roots. These <characteristic roots> determine growth, decay and oscillation of the linearized modes.
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