Characteristic equation

Definition

Let $a_0, \dots, a_{n-1}$ be real numbers, and consider the homogeneous linear equation with constant coefficients $$y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = 0.$$ Its characteristic polynomial is $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0$. Its characteristic equation is the polynomial equation $p(r) = 0$ for an unknown complex number $r$.

Remarks

For each complex $r$, the function $y = e^{r x}$ satisfies $y^{(n)} + \dots + a_0 y = p(r)\, e^{r x}$. So $e^{rx}$ is a solution if and only if $p(r) = 0$. Example: the equation $y'' - 3 y' + 2 y = 0$ has the characteristic equation $r^2 - 3 r + 2 = 0$, with roots $1$ and $2$. The characteristic polynomial of a linear recurrence plays the same role for recurrences, with $r^k$ in place of $e^{r x}$.

Explanations

L1 human

Try a solution of the form $e^{r x}$. Each derivative multiplies it by $r$, so the equation becomes $p(r) e^{rx} = 0$. Since $e^{rx}$ is never zero, $r$ must be a root of $p$.