Is the auxiliary equation of a differential equation related to characteristic polynomial for matrix eigenvalues?

Question

I am taking a course on differential equations and one of the topics is solving second order differentials with the help of an auxillary equation.

However one thing that's been bugging me alot is that I can't help but notice the similarity between say the auxiliary equation for a differential $ay'' + by' + cy=0$

$$a\lambda^2 + b\lambda + c=0$$

And the general form for the characteristic polynomial for a 2x2 matrix

$$a\lambda^2 + b\lambda + c=0$$

Does this happen to just be a coincidence or is there some link between the two concepts?

Answer 1

Let $x_1 := y$ and $x_2 := \dot y$. Hence,

$$\begin{bmatrix} \dot x_1\\ \dot x_2\end{bmatrix} = \begin{bmatrix} 0 & 1\\ -\frac{c}{a} & -\frac{b}{a}\end{bmatrix} \begin{bmatrix} x_1\\ x_2\end{bmatrix}$$

Compute the characteristic polynomial of the matrix above.