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I have heard it said that the Laplace transform has "many applications". But the only "applications" that I ever came across are those that require one to solve either some kind of differential equation.

Can you provide an application of the transform, where the transform of the function allows one to solve a problem, but which is not differential equation related.

For example, the Fourier transform has applications to signal processing, where one sends/receives signals by transforming and inverting them afterwards. This application with the Fourier transform has no interest in any differential equation solution. Is there a similar application for Laplace?

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    $\begingroup$ Control engineering and geophysics come to mind, as well as any situation where you can assume a continuous linear time invariant system: See this question on DSP for reference: dsp.stackexchange.com/questions/26146/… $\endgroup$ – Paul Apr 14 '16 at 21:38
  • $\begingroup$ The Fourier transform is just a special case of the Laplace transform, so your example actually works for both. I would argue that your example is still a case of solving a differential equation, even if you don't include the equal sign when you write the problem down on paper. $\endgroup$ – Chris Mueller Apr 15 '16 at 12:08
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I think you are neglecting the broad range of applications included in solving 'some kind of differential equation.' At the very basic level of physics, everything we know about the world is expressed in terms of differential equations. Some examples:

At the much more complex level of engineering, many of the problems are still expressed in terms of differential equations. To give you a feel, here are some questions from this site:

The Laplace transform, in particular, is used widely to 'solve some kind of differential equation' in these applications:

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