> convert Calculus derivative equations into algebraic ones...
This trick is - AFAIR from my classes - the basis of control theory math. You convert a control system to algebra, do your work there, and in the end, you convert back to differential equations.
I remember taking Control theory classes and thinking 'Why the hell did it take me 4-5 years to get to this level of mathematical mastery.'
It was amazing to be able to just do a little bit of odd math here or there, and suddenly have the solution on how to fix something completely unrelated, like how to tilt an airplane safely or make a robot balance a pen upright or something, by modeling it. Granted, we did very simple models, but the sheer scalability and power of these techniques is what made me feel on top of the world.
It should say Laplacian in OP's post, not Laplacian Transform (if I am not mistaken). The Laplacian is a matrix of (partial) derivitives and is used for the equational conversion.
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.
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).
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).
This trick is - AFAIR from my classes - the basis of control theory math. You convert a control system to algebra, do your work there, and in the end, you convert back to differential equations.