Right. Showing that is a matter of plugging things in and doing the usual integration by parts for the Laplace Transform.

I'm very puzzled by the second part. I don't buy it. Are you sure you've typed it up correctly? If you have the second-order ODE

then the solutions are either exponentials or hyperbolic trig functions (they're equivalent).

For simplicity, I'll write

Taking the derivative of this expression with respect to definitely does not give you

This is a very interesting problem, and very similar to the diffusion problem I did with my father for my senior seminar project at Grove City College. The fun part is doing the inverse Laplace Transform! You have to go to the line integral definition and use residue calculus: you get an infinite series solution.