I am self-studying a physics textbook on waves. While discussing solutions to linear homogeneous ODEs, the author talked about the exponential as "irreducible" solutions and on a footnote, said that they were, formally, "irreducible representations of the translation group". I am understanding the material that is presented in the book, but I think I would like to explore the relevant area of mathematics alongside. Which branch of mathematics is this and what are the introductory references?
Another persistent theme in the book is solving differential equations (oscillating systems, but a general approach would also be good for me) using arguments such as linearity and symmetry. What are good books on differential equations that approach the subject in a similar fashion and are accessible.
I have studied calculus and ODEs before, but it was very "application oriented" and most of the material was presented as "methods" (i.e you have a linear 1st order ODE, you use integrating factor, etc). I know with a lot of contemplation I can connect some dots myself, but it will be helpful to have some guiding material.