Laplace Transform of Composition of Functions

[Nocturne]

Hello,

I am trying to take the Laplace transform of floor(f(t)) in order to solve the differential equation f'=floor(f(t)). I know that L(floor(t)) = (e^(-s))/(s(1-e^(-s))) and that L(f(t)) = F(t) (of course), but I realized that I have no idea how to take the Laplace transform of a composition of functions, and no table I have been able to find contains L(floor(f(t))) or rules about compositions of functions. There is plenty of information on convolutions, but that isn't (as far as I can tell) what I'm looking for.

My question, at its essence, is this: given functions f and g, how do I determine L(f(g(t))? More specifically I want to know L(floor(f(t))), but any insight on the general case would be much appreciated.

I apologize if I am missing something obvious here, as well as for not knowing LaTeX.

Thank you!

 

[Nocturne]

I have re-posted my question in the differential equations forum https://www.physicsforums.com/showthread.php?p=2526341", as it seems more germane to that topic.

I originally posted in calculus/analysis because my query was essentially just about taking the Laplace transform itself, an integral operator, rather than actually solving the differential equation once I knew the transform. However, given the utility of the Laplace transform to the field of differential equations, I have changed my mind. Please post any advice or solutions to that thread instead.

Thank you!