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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:

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

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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?

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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.

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  • $\begingroup$ thanks for reply,just one question if there is given initial value and i use fourier transform,will my attempt fail to solve such problem? $\endgroup$ – dato datuashvili Dec 11 '13 at 19:36
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    $\begingroup$ @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. $\endgroup$ – Ron Gordon Dec 11 '13 at 19:57

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