# Fourier transform and Laplace transform to solve differential equation

generally we know that both Fourier transform and Laplace transform both is used to solve differential equation,first of all let us recall both form,first Fourier transform:

some times instead of $-2*\pi*f$,it is written $\omega$,and we know also Laplace transform

in some case low bound starts $-\infty$, and we know that

$s=\sigma+\omega*t$ ,which means that Fourier transform is a special case of Laplace when $\sigma=0$

but both methods is used to solve differential equation,my question us following how could i guess which method should i used in given concrete example?Laplace transform is used to transform differential equation into algebraic equation which simplify things,but what about Fourier transform?what it's goal in solving differential equation?

The goals of both transform techniques is to convert an ordinary differential equation into an algebraic one (and a partial differential equation into an ordinary differential equation). The difference is that the domain of the differential operators for which a Fourier tranform applies is $(-\infty,\infty)$, while that for which Laplace applies is $[0,\infty)$. Thus, LT's are used in initial value problems, while FT's are used when the function is applied over the whole real line.
• @datodatuashvili: it just may not make any sense. If you have an initial condition, then you have a "boundary" at $t=0$. Typically, the domain of such problems is $t \ge 0$, so FT would be unwieldy. – Ron Gordon Dec 11 '13 at 19:57