Can every differential equation be solved with the Laplace transformation??
so yeah , why?
You can rewrite any equation using the Laplace transform, but that doesn't mean you can always solve the integral exactly. The same holds for Fourier transforms, but most Physicists don't seem to understand that point.
-Dan (A Physicist that believes everything can be solved by taking a Fourier transform.)
As Topsquark said, you can apply the Laplace transform to both sides of any differential equation but it does not necessarily give anything useful. Except in very special cases, Laplace transforming a non-linear differential doesn't help at all. And even line equations, with non-constant coefficients, will be difficult at best. Frankly, I've never found an example where the "usual methods" (characteristic equation, etc.) was not simpler than the Laplace transform method. The only "advantage" to the Laplace transform method is that it gives very mechanical way to solve very simple d.e.s- Apply the Laplace transform, do some algebraic manipulation, and look the answer up in a table. That's the kind of thing engineers love and mathematicians hate.