Laplace of t*x''(t) Pulling the derivative through the integral

[Saladsamurai]

Homework Statement

I am working on solving the DE: t*x" + x' +t*x
by use of the Laplace transform. Now if we just look at the first term and its transform, we have

$$L[tx''(t)] = \int_0^\infty tx''e^{-st}\,dt = -\int_0^\infty \frac{d}{ds}\left(x''e^{-st}\right)\,dt$$


Now in the next step of my text, they say that we can pull the derivative through the integral if we assume that the unknown x(t) is "well-behaved enough." Can someone please explain to my feeble mind what that means and how that allows us to reverse the limit processes? I would like to be able to understand it well enough that I can apply it to future problems.

Thanks!

 

[fzero]

Someone that knows more analysis might be able to give a better answer, but I think that it's usually enough that $$x(t)$$ is sufficiently well-behaved that the integral

$$\int_0^\infty \left(x''e^{-st}\right)\,dt$$

exists and is a differentiable function of s.