in general, assuming that the solution has a Laplace transform, aren't we restricting ourselves to a subset of the solution set of the ODE? My professor in a lecture on ODEs, stated the following:

Consider the following ODE $$y'' - 2y = f(x),$$ where the Laplace
  transform of $f(x)$ exists.
To solve this ODE, we can argue that; Assume the solution of this ODE
  has a Laplace transform and by using the properties of LP,  $$L(y''
 -2y) = L(f(x))$$ [...] hence we get an algebraic equation instead of an DE.

I mean as far as I know this is a classical argument that is made while solving an ODE with the method of Laplace (or any other) transform method.
However, in general, assuming that the solution has a Laplace transform, aren't we restricting ourselves to a subset of the solution set of the ODE ?
I mean, in general, can't the ODE have a solution that does not have a Laplace transform ?

It's more clear how the first version generalizes to other DEs, by just factoring the relevant polynomial. Abstraction is useful...
