Maxima has a fairly serviceable Laplace transform utility built-in. Here’s an example from the popular ordinary differential equations book by Blanchard, Devaney and Hall:

Trouble arises when we look at discontinuous forcing functions, which is especially sad because it seems to me that’s what makes it worthwhile to spend time with Laplace transforms. In particular, the inverse transform function **ilt()** fails on the Heaviside Function and Dirac Delta, even though the built-in **laplace()** treats them correctly:

So, I’ve written an alternative inverse Laplace function **laplaceInv()** that fixes that problem:

Here are a few differential equation solutions to show how the new function behaves:

A second-order linear equation with a constant forcing function that vanishes at

A second-order equation with two impulsive forces:

The new Laplace transform Maxima function can be downloaded here.