Math 213 at uwaterloo has publicly available notes in spades: https://www.gabrielwong.net/notes/math213_notes.pdf#page9

I'm not sure this is what you're looking for, but the Feynman lectures give the basic idea in an elementary way: https://www.feynmanlectures.caltech.edu/I_23.html

> This idea of using exponentials in linear differential equations is almost as great as the invention of logarithms, in which multiplication is replaced by addition. Here differentiation is replaced by multiplication. . . . See how simple it is! Differential equations are immediately converted, by sight, into mere algebraic equations

Haven't read it but this currently-free Springer textbook covers DEs, Laplace and Fourier transformations, "Differential Equations and Their Applications": https://link.springer.com/book/10.1007%2F978-1-4612-4360-1

search for "Introduction to Signals & Systems"