Fourier transform will show up for an harmonic oscilator in the whole real line with incoming and outgoing wave boundary conditions, while Laplace will show up when working on semi-infinite interval whit initial conditions and proper convergence at infinity.
These are the most common, but not the only transforms one can build. There are also Melin and Hankel transforms, and by playing with the operator, the domain and the boundary conditions, we can construct the adequate transform for each given problem.
Spectral theory of DE’s is such a beautiful topic.
> 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
Similarly, the laplace transform is also a change of basis. But the basis it chooses is a very special one --- it's the eigenvectors of the differential operator. Note that
So e^(ax) literally an eigenvector of `(d/dx)`. And as we all know, going to the eigenbasis of a given operator/linear transform/matrix makes it easier to manipulate. The laplace transform is a change of basis that digonalizes the differential operator. This makes it easy to solve differentials.