Most people who deal with electronics have heard of the Fourier transform. That mathematical process makes it possible for computers to analyze sound, video, and it also offers critical math insights for tasks ranging from pattern matching to frequency synthesis. The Laplace transform is less familiar, even though it is a generalization of the Fourier transform. [Steve Bruntun] has a good explanation of the math behind the Laplace transform in a recent video that you can see below.
There are many applications for the Laplace transform, including transforming types of differential equations. This comes up often in electronics where you have time-varying components like inductors and capacitors. Instead of having to solve a differential equation, you can perform a Laplace, solve using common algebra, and then do a reverse transform to get the right answer. This is similar to how logarithms can take a harder problem — multiplication — and change it into a simpler addition problem, but on a much larger scale.