From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the intuition of what the Laplacian operator is doing(its a function that is doing something like analyzing a parameter of a point and comparing it to a "local averaging operation" of that parameter at all the neighboring points), but I wanted to ask something a little different:
If we were to think of solution space for a differential equation and the span our solutions can have depending on initial values, lets say in the form of a phase portraits, is there a Laplacian equivalent, and when you compare the two, how are they mapping to each other in an intuitive sense. What could you think of as the "highlight" in the Laplacian solution space and what would they correspond to in the solution space of our general solution. By highlight I mean there are certain distinct traits in phase portraits we use, and sources/sinks/assymptotes/arrow direction are "highlights" of a regular phase portrait. Also would the Laplace transformation of solution space of a solution be considered a vector space, or a subspace of one?
EDIT: Putting up an example was suggested, so how about y" + y = 0.
Its general solution is in the form y = c1cos(t) + c2sin(t) Solving with Laplace transformations: (s^2+1)Y(s) - sy(0) - y'(0)=0 general form should be something like: Y(s)= (sy(0)+y'(0))/(s^2+1)