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Altough above is a kind of answer, i can add some point;
Some differential equations arise during solving RLC cct, in this case instead of solving it in time domain, you can use Laplace domain and convert back in time domain. This is becasue in laplace domain, you need easier calculation compared to time domain.
Fourier is used in signal frequency analysis commonly.
But these are only very few example, so best way is buying a book and reading.
Yes, Laplace transform comes into picture while solving differential equations, also in the case of electronics circuits.
The Fourier transform comes into picture when one is interested in shape of pulses and needs to resolve it into its constituent sinusoidal equivalents that constitute a typical pulse.
Google for Fourier transform too.
yes i agree with emresel, Fourier is frequency analysis. Laplace too.
Fourier analysis usually used to solve problem in LTI system.
but Laplace can do something that Fourier can't do.
For example, Laplace can analyze unstable system which Fourier can't do..
Laplace can use for analyze the stability of system..
this is what i know..
i'm undergraduate student..i also learning about this..
so correct me if i'm wrong..
if you really know about the detail, Signals and Systems book by Allan V. Oppenheim provide all you need about Fourier and Laplace..it discuss from basic about Fourier, Fourier transform in continuous time and discrete time,Laplace transform and Z transform..
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