6
$\begingroup$

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely generated modules over a PID.

He states:

the space of solutions to a homogeneous linear differential equation with constant coefficients has a $k[D]$-module structure, where $D$ is differentiation, and moreover it's finitely-generated. So the structure theorem tells you what kind of decomposition to expect, and then you explicitly construct the corresponding generalized eigenvectors $x^ne^{rx}$.

I don't quite understand this, since I don't know what a $k[D]$-module is, and I'm not at all familiar with derivations.

I'm very interested in seeing explicitely how the description of the solutions of these equations is a special case of the structure theorem.

3 Answers 3