laplace_transform_1

Please expand to view my full discussion here.

I have created a sequence of videos that takes a non-standard approach to introducing the Laplace transform. The usual order is to introduce the transform through its definition, which involves an improper integral. Actual use of the transform, however, is much more akin to taking derivatives, which most students are far more more comfortable with, so I start with the" mechanical" process of finding and using transforms. The full development here goes as follows:

Videos 1 and 2 introduce the ideas of operators and transforms as black boxes that operate linearly, which is exactly how most of us find derivatives. The viewer should not be put off here by the seemingly simple-minded introduction. These two videos lay the conceptual groundwork for understanding what the Laplace transform is and does.

Videos 3 and 4 show how to use a table and the linearity properties to find Laplace transforms and inverse Laplace transforms. These are the fundamental operations that a person needs to be able to perform in order to solve problems using the Laplace transform.

Video 5 demonstrates finding an inverse Laplace transform using a partial fraction decomposition, and Video 6 shows how to solve a simple second order initial value problem using Laplace transforms.

After watching those six videos one should perhaps watch the three videos on deriving Laplace transforms of specific functions, which is where many texts and professors would begin their discussion of the topic.

One can then move on to Laplace transforms of step functions, piecewise defined functions and the Dirac delta "function," which is where solving without Laplace transforms is much more difficult or impossible.