It should be "Laplace transform". This video has nothing about Laplacians at all.

Are you thinking of the Jacobian matrix? I'm not sure what you mean by equational conversion. The Laplacian matrix is from graph theory and doesn't involve derivatives. The Laplacian operator involves differentiation, but is not a matrix.

Sorry, sorry. You are right, I am thinking of the Jacobian.

What does the comment mean then?

> "Also, apparently the Laplacian Transform can be used to convert Calculus derivative equations into algebraic ones..."

That's the primary reason to use the Laplace transform, as seen in the video. A derivative x'(t) gets transformed into a product (and an initial condition), s X(s) - x(0), and similar for higher derivatives, so a differential equation transforms into an algebraic equation, which can be solved by rearranging. This video assumed the initial conditions like x(0) = 0, and its notation was quite sloppy/confusing in places, as it didn't clearly distinguish the names of the two functions, x(t) and X(s).

One more point to sort this out. Is Laplacian always that modified adjacency matrix in graph theory, or does it mean something else as well?

That's the "Laplacian matrix". "Laplacian" as a noun usually refers to the differential operator, and "Laplacian" as an adjective is attached to quite a few things as well as the natrix (mostly developed or worked on by Laplace, or based on such).

> It should say Laplacian in OP's post, not Laplacian Transform (if I am not mistaken).

Neither. It's "Laplace Transform".

> The Laplacian is a matrix of (partial) derivitives and is used for the equational conversion.

Bar the name, this is called Operational Calculus[0].

[0]: https://en.wikipedia.org/wiki/Operational_calculus