TITLE: Motivating the Laplace transform definition
QUESTION [66 upvotes]: In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem.
The Laplace transform of a function $f(t)$, for $t \geq 0$ is defined by $\int_{0}^{\infty} f(t) e^{-st} dt$.
How to avoid looking at this definition as "magical"? How to somehow discover it from more basic definitions?

REPLY [5 votes]: There is in fact a very good paper of 2011 which addresses this question exactly:
The Laplace Transform: Motivating the Definition
by Howard Dwyer
Abstract:

Most undergraduate texts in ordinary
  differential equations (ODE) contain a
  chapter covering the Laplace transform
  which begins with the definition of
  the transform, followed by a sequence
  of theorems which establish the
  properties of the transform, followed
  by a number of examples. Many students
  accept the transform as a Gift From
  The Gods, but the better students will
  wonder how anyone could possibly have
  discovered/developed it. This article
  outlines a presentation, which offers
  a plausible (hopefully) progression of
  thoughts, which leads to integral
  transforms in general, and the Laplace
  transform in particular.