I'm learning about laplace tranform method to solve lineair differential equations but i'm wondering if laplace transformations can be used to solve every linear differential equations there is. Or are there some limitations?
I know that for the operator method the equation has to be from the form: $e^at$ / $sin(bt)$ / $cos(bt)$ and Polynomial.
On
One can say without any problem that almost no equation can be solved using Laplace transforms. There are two main reasons for this. One is that in general Laplace transforms cannot be computed, unless for functions with some prescribed growth. Otherwise nothing can be done. Even worse, one often writes a symbol for the Laplace transform of the solution without knowing whether the solution has that prescribed growth (still one can say that if at the hand we find a solution of which one can compute the Laplace transform, the reasoning is correct). The other main reason is of course that the method expects that we are able to compute explicitly, and like finding primitives this is more art than mathematics.
A minor secondary reason is that the method has difficulties dealing easily with the situation when the initial conditions are not numbers...
I would say that there is absolutely no reason why we should use the method of Laplace transforms, in general for each problem there are much better methods.
Consider a function without a Laplace transform, such as $e^{e^x}$. Then:
$$y'(x)=e^{e^x}$$
can't be solved by taking Laplace transforms.