Kirchoff's Law for circuits is the most obvious answer... but other than that, I haven't been able to find any examples where Laplace Transforms can be used to solve a model of interesting real life examples.
If anyone has any examples of some ideas to look up, or any links/books to read on this area, then that would be great.
Thanks in advance.
First thing to note here:
So basically its unifying two different branches of mathematics. As far as I have known (I am not a Physics guy) L.T. is applied to circuit studies as you have mentioned. Like Fourier Transform, L.T. is also used in signal processing. For a more detailed reference you can check this out.