Characteristic equation of a constant-coefficient differential equation ($P(r)=0$)

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 $P(r)={\sum }_j{a}_j{r}^j=0$. A simple root gives an exponential solution; a root of multiplicity $m$ gives $x^k{e}^{rx}$ 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.