Laplace Transform I

As a part of my study, I looked up several applications of the Laplace Transformation Method in solving constant coefficient differential equations. The method is very straightforward and it beautifully exploits the property of the transform. In short, it converts differential system into algebraic system which can be solved by simple algebraic methods. Yet I'm wondering what is the physical meaning of transformation and no one whom I asked about this so far was able to give me a clear answer, which is quite unfortunate. Many people at various internet physics and mathematics forums are having basically the same question but without any answers. Is there truly no physical representation of the transform or are we yet to discover it?

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