In this post I’m going to discuss a topic that I’m currently covering in UW – La Crosse’s MTH 353 Differential Equations course: Laplace transforms. While (like many math topics) I didn’t appreciate transforms as much as I should have when I learned them for the first time, they have become my favorite topic to teach in the undergraduate curriculum.

First, Pierre-Simon Laplace was an absolute crusher in the mathematical sciences. In addition to the transform method that bears his name, he’s responsible for a lot of the theoretical underpinnings of Bayesian statistics (one of Richard’s favorite topics), tidal flow, spherical harmonics, potential theory and Laplace’s equation, among many other things.

The Laplace transform, in its simplest application, transforms linear, generally inhomogeneous, constant-coefficient ordinary differential equations of time t into an algebraic equation of a (complex) frequency variable s.

What I love the most about Laplace transforms as a topic in the mathematics curriculum is that is requires students to apply techniques from earlier in their training. For example:

– Completing the square (elementary algebra)

– Horizontal translation of functions (elementary algebra)

– Improper integration (second-semester calculus)

– Partial fraction decomposition (second-semester calculus)

– Linear transformations (linear algebra/functional analysis)

– Elementary theory of linear ODEs (elementary differential equations)

Another nice thing about the Laplace transform is that it can handle discontinuous inhomogeneous (forcing) data like Heaviside step functions and Dirac delta functions, as well as forcing terms that aren’t their own derivatives like polynomials, sines, cosines and exponential functions. When viewing the solution to differential equations in this setting, in the space of functions of the variable s, one can clearly see how initial data and forcing data is propagated in time back in the original solution space.

If you’re interested in Laplace transforms, I’ve created some videos for my MTH 353 course, and they can be found here on my YouTube page!