TITLE: Differential equations and Fourier and Laplace transforms
QUESTION [42 upvotes]: Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and some other situations where the Laplace transform is used instead.
Edit: In response to the comments. I would like to hear an answer in the context of pure mathematics. For example, I hear that the Fourier transform is very very useful in the theory of partial differential equations because it transforms a PDE into an algebraic equation. However, I don't hear about the Laplace transform being so useful in pure mathematics.

REPLY [3 votes]: Summarizing answers from The Straight Dope:
Fourier transform:

Imaginary argument (special case of Laplace transform)
Useful for seeing frequency spectrum of stable signals or functions converge to $0$ at infinity

Laplace tranform:

Real or complex argument
Useful in solving rate equations, work for expotentially diverging functions (i.e. checking their stability)