I know it seems like an weird question but, it seems a little odd that one thing in math is able to do two completely unrelated things. Unless of course the two things are related somehow?
So I have to ask, what does decomposing a function into its sinusoids have to do with solving differential equations(or more specifically converting a differential equation to an algebraic one)?
The question extends to the Laplace transforms as well but, I feel the answer to both are the same.
Sidenote: If you think you understand what I'm asking and you feel you can phrase it better, please comment below. I'll update my question.
On $L^2$, the Fourier transform $\mathcal F$ is a unitary operator that "diagonalizes" the derivative operator $D$. $$ \mathcal F(D \phi)(\xi) = 2\pi i\xi \;\mathcal F(\phi)(\xi) $$ Of course working with a diagonal matrix (and here a multiplication operator) is easier than a general matrix, right?