Ever since I learned about the Laplace transform method in undergraduate engineering, I’ve always been somewhat mystified by it. The Laplace transform of a function is defined by:
The Wikipedia page has all of the information on what the Laplace transform is used for and why it is useful. Briefly, a Laplace transform allows for easy solution of -order linear non-homogeneous differential equations, as well as some partial differential equations. However, the Wiki is somewhat light on the history and origins of the transform.
My question is: what motivated Laplace to dream up such a strange integral, and apply it to solving differential equations?
This source says it was first used by Laplace to prove the central limit theorem.
Arthur Mattuck has given a pretty good lecture connecting the Laplace Transform with the power series, but I still don’t know if that’s what Laplace was thinking when he made his discovery. Unfortunately, this source probably has all the information I’d like in it, but it is paywalled.