Characteristic polynomial

The characteristic polynomial for a linear ODE is a polynomial of $\lambda$ where the derivatives of dependent variable are substituted for powers of $\lambda$. E.g. the characteristic polynomial for $x^{(3)} + e^t \ddot x + \cos(t) \dot x + 43x = 7$ is $\lambda^3 + e^t \lambda^2 + \cos(t) \lambda + 43 = 0$.

If we consider $D = \frac{d}{dx}$ to be the differential operator and if $p(\lambda)$ is our characteristic polynomial, then we can write the ODE above equivalently as $p(D)x=0$.

Solving the characteristic polynomial is helpful in solving the corresponding differential equation (see linear ODEs).