I've seen in a loit of courses that we just take the transform of an equation, manipulate/solve it and go back having found a solution. The best example would be the heat equation. In Wikipedia it is illustrated another example in which if the symbol is positive then by solving the transform I get the solution.
To me it all seem to be a trick. Do we know that we can do it or are we just trying and getting lucky? For example this trick doesn't work for the Laplace equation. Even though in that case we get that the solution is a polynomial, so we still get something, but far less than we would like.
My question is: when are we sure that we can use this trick? When can't we use it?